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<div class="add">

<span class="kw">set </span><font color="Maroon" title="c3">TM</font> =  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">G</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E2:1_1_1"/><span class="lab"><font color="Green" title="E4">A4</font></span>: 
<font color="Maroon" title="c4">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font>
 <span class="kw">and </span><br/><a NAME="E3:1_1_1"/><span class="lab"><font color="Green" title="E5">A5</font></span>: 
<font color="Maroon" title="c4">G</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:1_1_1"/><span class="lab"><font color="Green" title="E6">A6</font></span>: 
<font color="Maroon" title="c4">G</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E1">A1</font></span></a>, <a class="txt" href="#E2:1"><span class="lab"><font color="Green" title="E2">A2</font></span></a>, <a class="txt" href="#E3:1"><span class="lab"><font color="Green" title="E3">A3</font></span></a>, <a class="ref" href="compts_1.html#D1" title="COMPTS_1:def.1">COMPTS_1:def 1</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_1"/>
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> or  not  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_1"/><span class="lab"><font color="Green" title="E7">A7</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/> for <font color="Olive" title="b2">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>  ex <font color="Olive" title="b3">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> st <br/>( <font color="Olive" title="b3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Olive" title="b2">p</font>,<font color="Olive">b<sub>1</sub></font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">A</font> )</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c5">R</font> = 1; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>  ex <font color="Olive" title="b2">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> st <br/>( <font color="Olive" title="b2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c5">R</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">A</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> st <br/>( <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c5">R</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> )</span><br/><span class="kw">take </span>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c5">R</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> )</span><br/><span class="kw">thus </span><a NAME="E2:1_1_1_1_1"/>
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c5">R</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1"><span class="lab"><font color="Green" title="E7">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_2"/><span class="lab"><font color="Green" title="E7">A8</font></span>: 
 not  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/> for <font color="Olive" title="b2">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>  ex <font color="Olive" title="b3">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> st <br/>( <font color="Olive" title="b3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Olive" title="b2">p</font>,<font color="Olive">b<sub>1</sub></font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">A</font> )</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c5">cTM</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c6">FUNCS</font> =  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>);<br/><span class="kw">consider </span><font color="Maroon" title="c7">g</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E3:1_1_1_1_2"/><span class="lab"><font color="Green" title="E8">A9</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font>
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_1_2"/>
<font color="Maroon" title="c7">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1"><span class="lab"><font color="Green" title="E6">A6</font></span></a>, <a class="ref" href="finseq_4.html#T58" title="FINSEQ_4:th.58">FINSEQ_4:58</a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  st <font color="Olive" title="b1">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; <font color="Olive" title="b2">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $3 &amp; <font color="Olive" title="b1">f1</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  holds <br/>( <font color="Olive" title="b2">f2</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp;  ex <font color="Olive" title="b3">A</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> st <br/>( <font color="Olive" title="b3">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> &amp; (  for <font color="Olive" title="b4">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> holds  <font color="Olive" title="b2">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">x</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist_min</a> <font color="Olive" title="b3">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">x</font></span>)</span>) ) ) );<br/><a NAME="E5:1_1_1_1_2"/>
 not  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">G</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1"><span class="lab"><font color="Green" title="E5">A5</font></span></a>, <a class="ref" href="setfam_1.html#T45" title="SETFAM_1:th.45">SETFAM_1:45</a></span>;<br/><a NAME="E6:1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A10</font></span>: 
 not <font color="Maroon" title="c7">g</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_1_2"><span class="lab"><font color="Green" title="E8">A9</font></span></a>, <a class="ref" href="zfmisc_1.html#T2" title="ZFMISC_1:th.2">ZFMISC_1:2</a></span>;<br/><a NAME="E7:1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A11</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">g</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/><a NAME="E8:1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A12</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E9:1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A13</font></span>: 
<font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">g1</font> = <font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_1_2"><span class="lab"><font color="Green" title="E8">A9</font></span></a></span>;<br/><a NAME="E11:1_1_1_1_2"/><span class="lab"><font color="Green" title="E13">A14</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">g1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_2"><span class="lab"><font color="Green" title="E7">A8</font></span></a>, <a class="txt" href="#E3:1_1_1_1_2"><span class="lab"><font color="Green" title="E8">A9</font></span></a>, <a class="txt" href="#E8:1_1_1_1_2"><span class="lab"><font color="Green" title="E11">A12</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E12:1_1_1_1_2"/>
<font color="Maroon" title="c8">g1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1"><span class="lab"><font color="Green" title="E2">A2</font></span></a>, <a class="txt" href="#E2:1_1_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a>, <a class="txt" href="#E3:1_1_1_1_2"><span class="lab"><font color="Green" title="E8">A9</font></span></a>, <a class="txt" href="#E9:1_1_1_1_2"><span class="lab"><font color="Green" title="E12">A13</font></span></a>, <a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">g19</font> = <font color="Maroon" title="c8">g1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:1_1_1_1_2"><span class="lab"><font color="Green" title="E13">A14</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">Dg19</font> =  <a href="weierstr.html#K6" title="WEIERSTR:func.6">dist_min</a> <font color="Maroon" title="c9">g19</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>) <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T8" title="FUNCT_2:th.8">FUNCT_2:8</a>, <a class="ref" href="topmetr.html#T17" title="TOPMETR:th.17">TOPMETR:17</a></span>;<br/><a NAME="E15:1_1_1_1_2"/><span class="lab"><font color="Green" title="E14">A15</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">g</font> holds <br/> for <font color="Olive" title="b2">x</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)  ex <font color="Olive" title="b3">y</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>) st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> &amp; <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">g</font> implies  for <font color="Olive" title="b1">x</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)  ex <font color="Olive" title="b2">y</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>) st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">n</font>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_1_1_2_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font>
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E15">A16</font></span>: 
<font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)  ex <font color="Olive" title="b2">y</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>) st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">n</font>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>) st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c13">fx</font> = <font color="Maroon" title="c12">x</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T17" title="TOPMETR:th.17">TOPMETR:17</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_1_2_1_1"/>
(  not <font color="Maroon" title="c13">fx</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  or <font color="Maroon" title="c13">fx</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E16">A17</font></span>: 
 not <font color="Maroon" title="c13">fx</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>) st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c14">y</font> = <font color="Maroon" title="c12">x</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c14">y</font>]</span><br/><span class="kw">thus </span><a NAME="E2:1_1_1_1_2_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c14">y</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E16">A17</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E16">A18</font></span>: 
<font color="Maroon" title="c13">fx</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>) st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><a NAME="E2:1_1_1_1_2_1_1_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 &amp; <font color="Maroon" title="c11">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">g</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_1_2_1"><span class="lab"><font color="Green" title="E15">A16</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:1_1_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A19</font></span>: 
<font color="Maroon" title="c11">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E4:1_1_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_1_2"><span class="lab"><font color="Green" title="E8">A9</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">A</font> = <font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> ;<br/><a NAME="E6:1_1_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_2"><span class="lab"><font color="Green" title="E7">A8</font></span></a>, <a class="txt" href="#E3:1_1_1_1_2"><span class="lab"><font color="Green" title="E8">A9</font></span></a>, <a class="txt" href="#E3:1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E17">A19</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">A9</font> = <font color="Maroon" title="c14">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> ;<br/><a NAME="E8:1_1_1_1_2_1_1_2"/>
 <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of  <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T2" title="TSEP_1:th.2">TSEP_1:2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">h</font> being   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a><span class="kw"> such that </span><br/><a NAME="E10:1_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E18">A20</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> holds  <font color="Maroon" title="c16">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c13">fx</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"><span class="p2">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist_min</a> <font color="Maroon" title="c15">A9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="ref" href="topgen_5.html#T50" title="TOPGEN_5:th.50">TOPGEN_5:50</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">y</font> = <font color="Maroon" title="c16">h</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>) <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T8" title="FUNCT_2:th.8">FUNCT_2:8</a>, <a class="ref" href="topmetr.html#T17" title="TOPMETR:th.17">TOPMETR:17</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c17">y</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c17">y</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c18">f1</font>, <font color="Maroon" title="c19">f2</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font> &amp; <font color="Maroon" title="c19">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y</font> &amp; <font color="Maroon" title="c18">f1</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  implies ( <font color="Maroon" title="c19">f2</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp;  ex <font color="Olive" title="b1">A</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> st <br/>( <font color="Olive" title="b1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> &amp; (  for <font color="Olive" title="b2">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> holds  <font color="Maroon" title="c19">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist_min</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font></span>)</span>) ) ) ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E12:1_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E19">A21</font></span>: 
<font color="Maroon" title="c18">f1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c12">x</font>
 <span class="kw">and </span><br/><a NAME="E13:1_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E20">A22</font></span>: 
<font color="Maroon" title="c19">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c17">y</font>
 <span class="kw">and </span><br/><a NAME="E14:1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c18">f1</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c19">f2</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp;  ex <font color="Olive" title="b1">A</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> st <br/>( <font color="Olive" title="b1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> &amp; (  for <font color="Olive" title="b2">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> holds  <font color="Maroon" title="c19">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist_min</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font></span>)</span>) ) ) )</span><br/><span class="kw">thus </span><a NAME="E15:1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c19">f2</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E20">A22</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">A</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> st <br/>( <font color="Olive" title="b1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> &amp; (  for <font color="Olive" title="b2">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> holds  <font color="Maroon" title="c19">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist_min</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font></span>)</span>) ) )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c15">A9</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">A9</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> &amp; (  for <font color="Olive" title="b1">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> holds  <font color="Maroon" title="c19">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">f1</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"><span class="p2">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist_min</a> <font color="Maroon" title="c15">A9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span>) ) )</span><br/><span class="kw">thus </span><a NAME="E16:1_1_1_1_2_1_1_2"/>
( <font color="Maroon" title="c15">A9</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> &amp; (  for <font color="Olive" title="b1">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> holds  <font color="Maroon" title="c19">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">f1</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"><span class="p2">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist_min</a> <font color="Maroon" title="c15">A9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E12:1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E13:1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E20">A22</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c11">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)<span class="kw"> such that </span><br/><a NAME="E17:1_1_1_1_2"/><span class="lab"><font color="Green" title="E15">A23</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">g</font>
 <span class="kw">and </span><br/><a NAME="E18:1_1_1_1_2"/><span class="lab"><font color="Green" title="E16">A24</font></span>: 
( <font color="Maroon" title="c11">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Dg19</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 <span class="kw">and </span><br/><a NAME="E19:1_1_1_1_2"/><span class="lab"><font color="Green" title="E17">A25</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">g</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c11">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c11">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_2.html#S1" title="NAT_2:sch.1">NAT_2:sch 1</a>(<a class="txt" href="#E15:1_1_1_1_2"><span class="lab"><font color="Green" title="E14">A15</font></span></a>);<br/></span><a NAME="E20:1_1_1_1_2"/><span class="lab"><font color="Green" title="E18">A26</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_1_1_2"><span class="lab"><font color="Green" title="E10">A11</font></span></a>, <a class="txt" href="#E17:1_1_1_1_2"><span class="lab"><font color="Green" title="E15">A23</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E21:1_1_1_1_2"/>
<font color="Maroon" title="c11">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">p</font></span>)</span> is    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">pL</font> = <font color="Maroon" title="c11">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">p</font></span>)</span> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T17" title="TOPMETR:th.17">TOPMETR:17</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">rngPL</font> =  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c12">pL</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c14">cRpL</font> =  <a href="weierstr.html#K1" title="WEIERSTR:func.1">[#]</a> <font color="Maroon" title="c13">rngPL</font>;<br/><a NAME="E24:1_1_1_1_2"/><span class="lab"><font color="Green" title="E19">A27</font></span>: 
 <a href="weierstr.html#K1" title="WEIERSTR:func.1">[#]</a> <font color="Maroon" title="c13">rngPL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c12">pL</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="weierstr.html#D1" title="WEIERSTR:def.1">WEIERSTR:def 1</a></span>;<br/><a NAME="E25:1_1_1_1_2"/><span class="lab"><font color="Green" title="E20">A28</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:1_1_1_1_2"><span class="lab"><font color="Green" title="E15">A23</font></span></a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  st $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">p</font> &amp; <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1 holds <br/>( <font color="Olive" title="b1">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; (  for <font color="Olive" title="b2">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/>  for <font color="Olive" title="b3">h</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  st <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">p</font> &amp; <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">j</font> holds <br/> for <font color="Olive" title="b4">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> holds  <font color="Olive" title="b3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">x</font> ) );<br/><a NAME="E26:1_1_1_1_2"/><span class="lab"><font color="Green" title="E21">A29</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">n</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E22">A30</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c16">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">p</font> &amp; <font color="Maroon" title="c16">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> implies ( <font color="Maroon" title="c16">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; (  for <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/>  for <font color="Olive" title="b2">h</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  st <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 &amp; <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">p</font> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font> holds <br/> for <font color="Olive" title="b3">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> holds  <font color="Olive" title="b2">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x</font> ) ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E23">A31</font></span>: 
<font color="Maroon" title="c15">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">p</font>
 <span class="kw">and </span><br/><a NAME="E3:1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E24">A32</font></span>: 
<font color="Maroon" title="c16">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; (  for <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/>  for <font color="Olive" title="b2">h</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  st <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 &amp; <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">p</font> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font> holds <br/> for <font color="Olive" title="b3">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> holds  <font color="Olive" title="b2">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x</font> ) )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_2_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_1_2_2_1"/>
( <font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or <font color="Maroon" title="c15">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E25">A33</font></span>: 
<font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; (  for <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/>  for <font color="Olive" title="b2">h</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  st <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 &amp; <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">p</font> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font> holds <br/> for <font color="Olive" title="b3">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> holds  <font color="Olive" title="b2">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x</font> ) )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_1_2_2_1_1"/>
<font color="Maroon" title="c16">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_1_1_2"><span class="lab"><font color="Green" title="E9">A10</font></span></a>, <a class="txt" href="#E18:1_1_1_1_2"><span class="lab"><font color="Green" title="E16">A24</font></span></a>, <a class="txt" href="#E3:1_1_1_1_2_2"><span class="lab"><font color="Green" title="E24">A32</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/>  for <font color="Olive" title="b2">h</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  st <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 &amp; <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">p</font> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font> holds <br/> for <font color="Olive" title="b3">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> holds  <font color="Olive" title="b2">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c17">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">h</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  st <font color="Maroon" title="c17">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 &amp; <font color="Maroon" title="c17">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">p</font> &amp; <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">j</font> holds <br/> for <font color="Olive" title="b2">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> holds  <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c18">g</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 &amp; <font color="Maroon" title="c17">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">p</font> &amp; <font color="Maroon" title="c18">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">j</font> implies  for <font color="Olive" title="b1">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> holds  <font color="Maroon" title="c18">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:1_1_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E26">A34</font></span>: 
<font color="Maroon" title="c17">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E27">A35</font></span>: 
<font color="Maroon" title="c17">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">p</font>
 <span class="kw">and </span><br/><a NAME="E5:1_1_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E28">A36</font></span>: 
<font color="Maroon" title="c18">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> holds  <font color="Maroon" title="c18">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span><br/><a NAME="E6:1_1_1_1_2_2_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E27">A35</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><span class="kw">hence </span><a NAME="E7:1_1_1_1_2_2_1_1"/>
 for <font color="Olive" title="b1">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> holds  <font color="Maroon" title="c18">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_1_2_2"><span class="lab"><font color="Green" title="E24">A32</font></span></a>, <a class="txt" href="#E1:1_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E25">A33</font></span></a>, <a class="txt" href="#E3:1_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E26">A34</font></span></a>, <a class="txt" href="#E5:1_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E28">A36</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_2_2_1_2"/>
<font color="Maroon" title="c15">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; (  for <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/>  for <font color="Olive" title="b2">h</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  st <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 &amp; <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">p</font> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font> holds <br/> for <font color="Olive" title="b3">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> holds  <font color="Olive" title="b2">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x</font> ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">n1</font> = <font color="Maroon" title="c15">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T20" title="NAT_1:th.20">NAT_1:20</a></span>;<br/><a NAME="E3:1_1_1_1_2_2_1_2"/>
<font color="Maroon" title="c11">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">n</font> is    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">pn</font> = <font color="Maroon" title="c11">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">n</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T17" title="TOPMETR:th.17">TOPMETR:17</a></span>;<br/><a NAME="E5:1_1_1_1_2_2_1_2"/>
<font color="Maroon" title="c15">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_1_2_2"><span class="lab"><font color="Green" title="E23">A31</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E6:1_1_1_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A37</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">n1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 &amp; <font color="Maroon" title="c15">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E7:1_1_1_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A38</font></span>: 
<font color="Maroon" title="c15">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E8:1_1_1_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A39</font></span>: 
<font color="Maroon" title="c18">pn</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_2_2"><span class="lab"><font color="Green" title="E22">A30</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:1_1_1_1_2_2_1_2"/>
<font color="Maroon" title="c16">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:1_1_1_1_2"><span class="lab"><font color="Green" title="E15">A23</font></span></a>, <a class="txt" href="#E19:1_1_1_1_2"><span class="lab"><font color="Green" title="E17">A25</font></span></a>, <a class="txt" href="#E3:1_1_1_1_2_2"><span class="lab"><font color="Green" title="E24">A32</font></span></a>, <a class="txt" href="#E6:1_1_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E25">A37</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/>  for <font color="Olive" title="b2">h</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  st <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 &amp; <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">p</font> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font> holds <br/> for <font color="Olive" title="b3">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> holds  <font color="Olive" title="b2">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x</font></span><br/><span class="kw">consider </span><font color="Maroon" title="c19">A</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E11:1_1_1_1_2_2_1_2"/>
<font color="Maroon" title="c19">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E12:1_1_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E28">A40</font></span>: 
 for <font color="Olive" title="b1">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> holds  <font color="Maroon" title="c16">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">pn</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist_min</a> <font color="Maroon" title="c19">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:1_1_1_1_2"><span class="lab"><font color="Green" title="E15">A23</font></span></a>, <a class="txt" href="#E19:1_1_1_1_2"><span class="lab"><font color="Green" title="E17">A25</font></span></a>, <a class="txt" href="#E3:1_1_1_1_2_2"><span class="lab"><font color="Green" title="E24">A32</font></span></a>, <a class="txt" href="#E6:1_1_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E25">A37</font></span></a>, <a class="txt" href="#E8:1_1_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E27">A39</font></span></a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c20">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">h</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  st <font color="Maroon" title="c20">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 &amp; <font color="Maroon" title="c20">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">p</font> &amp; <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">j</font> holds <br/> for <font color="Olive" title="b2">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> holds  <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c21">h</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c20">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 &amp; <font color="Maroon" title="c20">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">p</font> &amp; <font color="Maroon" title="c21">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">j</font> implies  for <font color="Olive" title="b1">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> holds  <font color="Maroon" title="c21">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E13:1_1_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E29">A41</font></span>: 
<font color="Maroon" title="c20">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 <span class="kw">and </span><br/><a NAME="E14:1_1_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E30">A42</font></span>: 
<font color="Maroon" title="c20">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">p</font>
 <span class="kw">and </span><br/><a NAME="E15:1_1_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E31">A43</font></span>: 
<font color="Maroon" title="c21">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> holds  <font color="Maroon" title="c21">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c22">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c21">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font></span><br/><a NAME="E16:1_1_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E32">A44</font></span>: 
<font color="Maroon" title="c16">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">pn</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist_min</a> <font color="Maroon" title="c19">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:1_1_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E28">A40</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_2_2_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_1_2_2_1_2_1"/>
( <font color="Maroon" title="c20">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">n</font> or <font color="Maroon" title="c20">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:1_1_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E29">A41</font></span></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_2_2_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_2_2_1_2_1_1"/><span class="lab"><font color="Green" title="E33">A45</font></span>: 
<font color="Maroon" title="c20">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c21">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font></span><br/><div class="add"><a NAME="E2:1_1_1_1_2_2_1_2_1_1"/><span class="lab"><font color="Green" title="E34">A46</font></span>: 
<font color="Maroon" title="c18">pn</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E32">A44</font></span></a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/><a NAME="E3:1_1_1_1_2_2_1_2_1_1"/>
<font color="Maroon" title="c21">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">pn</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_2_2"><span class="lab"><font color="Green" title="E22">A30</font></span></a>, <a class="txt" href="#E7:1_1_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E26">A38</font></span></a>, <a class="txt" href="#E14:1_1_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E30">A42</font></span></a>, <a class="txt" href="#E15:1_1_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E31">A43</font></span></a>, <a class="txt" href="#E1:1_1_1_1_2_2_1_2_1_1"><span class="lab"><font color="Green" title="E33">A45</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_1_2_2_1_2_1_1"/>
<font color="Maroon" title="c21">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_1_2_2_1_2_1_1"><span class="lab"><font color="Green" title="E34">A46</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_2_2_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_2_2_1_2_1_2"/>
<font color="Maroon" title="c20">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c21">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_1_2_2_1_2_1_2"/>
<font color="Maroon" title="c21">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_1_2_2"><span class="lab"><font color="Green" title="E24">A32</font></span></a>, <a class="txt" href="#E15:1_1_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E31">A43</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E27:1_1_1_1_2"/><span class="lab"><font color="Green" title="E22">A47</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E28:1_1_1_1_2"/><span class="lab"><font color="Green" title="E23">A48</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E27:1_1_1_1_2"><span class="lab"><font color="Green" title="E22">A47</font></span></a>, <a class="txt" href="#E26:1_1_1_1_2"><span class="lab"><font color="Green" title="E21">A29</font></span></a>);<br/></span><a NAME="E29:1_1_1_1_2"/><span class="lab"><font color="Green" title="E24">A49</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">pL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E30:1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">pL</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c12">pL</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T113" title="RELAT_1:th.113">RELAT_1:113</a></span>;<br/><a NAME="E31:1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A50</font></span>: 
<font color="Maroon" title="c13">rngPL</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E1">A1</font></span></a>, <a class="txt" href="#E20:1_1_1_1_2"><span class="lab"><font color="Green" title="E18">A26</font></span></a>, <a class="txt" href="#E28:1_1_1_1_2"><span class="lab"><font color="Green" title="E23">A48</font></span></a>, <a class="ref" href="weierstr.html#T9" title="WEIERSTR:th.9">WEIERSTR:9</a></span>;<br/><a NAME="E32:1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A51</font></span>: 
 <a href="weierstr.html#K1" title="WEIERSTR:func.1">[#]</a> <font color="Maroon" title="c13">rngPL</font> is  <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="weierstr.html#T13" title="WEIERSTR:th.13">WEIERSTR:13</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">cRpL</font> =  <a href="weierstr.html#K1" title="WEIERSTR:func.1">[#]</a> <font color="Maroon" title="c13">rngPL</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:1_1_1_1_2"><span class="lab"><font color="Green" title="E25">A50</font></span></a>, <a class="ref" href="weierstr.html#T11" title="WEIERSTR:th.11">WEIERSTR:11</a>, <a class="ref" href="weierstr.html#D1" title="WEIERSTR:def.1">WEIERSTR:def 1</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c16">L</font> =  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c15">cRpL</font>;<br/><a NAME="E34:1_1_1_1_2"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c15">cRpL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">cRpL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:1_1_1_1_2"><span class="lab"><font color="Green" title="E26">A51</font></span></a>, <a class="ref" href="rcomp_1.html#T14" title="RCOMP_1:th.14">RCOMP_1:14</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E36:1_1_1_1_2"/><span class="lab"><font color="Green" title="E27">A52</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">pL</font>
 <span class="kw">and </span><br/><a NAME="E37:1_1_1_1_2"/><span class="lab"><font color="Green" title="E28">A53</font></span>: 
<font color="Maroon" title="c12">pL</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c15">cRpL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:1_1_1_1_2"><span class="lab"><font color="Green" title="E19">A27</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E38:1_1_1_1_2"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1"><span class="lab"><font color="Green" title="E5">A5</font></span></a>, <a class="ref" href="setfam_1.html#T45" title="SETFAM_1:th.45">SETFAM_1:45</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E40:1_1_1_1_2"/><span class="lab"><font color="Green" title="E29">A54</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Y</font>
 <span class="kw">and </span><br/><a NAME="E41:1_1_1_1_2"/><span class="lab"><font color="Green" title="E30">A55</font></span>: 
<font color="Maroon" title="c18">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_1_2"><span class="lab"><font color="Green" title="E8">A9</font></span></a>, <a class="txt" href="#E36:1_1_1_1_2"><span class="lab"><font color="Green" title="E27">A52</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">x</font> = <font color="Maroon" title="c17">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:1_1_1_1_2"><span class="lab"><font color="Green" title="E27">A52</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c20">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E44:1_1_1_1_2"/><span class="lab"><font color="Green" title="E31">A56</font></span>: 
<font color="Maroon" title="c20">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">g</font>
 <span class="kw">and </span><br/><a NAME="E45:1_1_1_1_2"/><span class="lab"><font color="Green" title="E32">A57</font></span>: 
<font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:1_1_1_1_2"><span class="lab"><font color="Green" title="E30">A55</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">j</font> = <font color="Maroon" title="c20">j</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:1_1_1_1_2"><span class="lab"><font color="Green" title="E31">A56</font></span></a></span>;<br/><a NAME="E47:1_1_1_1_2"/><span class="lab"><font color="Green" title="E33">A58</font></span>: 
<font color="Maroon" title="c21">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:1_1_1_1_2"><span class="lab"><font color="Green" title="E31">A56</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E48:1_1_1_1_2"/><span class="lab"><font color="Green" title="E34">A59</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:1_1_1_1_2"><span class="lab"><font color="Green" title="E31">A56</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_2_3"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_2_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_1_2_3_1"/>
( <font color="Maroon" title="c21">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c21">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:1_1_1_1_2"><span class="lab"><font color="Green" title="E34">A59</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_2_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_2_3_1_1"/><span class="lab"><font color="Green" title="E35">A60</font></span>: 
<font color="Maroon" title="c21">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c15">cRpL</font></span><br/><div class="add"><a NAME="E2:1_1_1_1_2_3_1_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">g19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1_1_1_2"><span class="lab"><font color="Green" title="E29">A54</font></span></a>, <a class="txt" href="#E45:1_1_1_1_2"><span class="lab"><font color="Green" title="E32">A57</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E3:1_1_1_1_2_3_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist_min</a> <font color="Maroon" title="c9">g19</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hausdorf.html#T9" title="HAUSDORF:th.9">HAUSDORF:9</a></span>;<br/><a NAME="E4:1_1_1_1_2_3_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist_min</a> <font color="Maroon" title="c9">g19</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1k.html#T9" title="JORDAN1K:th.9">JORDAN1K:9</a></span>;<br/><span class="kw">hence </span><a NAME="E5:1_1_1_1_2_3_1_1"/>
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c15">cRpL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:1_1_1_1_2"><span class="lab"><font color="Green" title="E15">A23</font></span></a>, <a class="txt" href="#E18:1_1_1_1_2"><span class="lab"><font color="Green" title="E16">A24</font></span></a>, <a class="txt" href="#E20:1_1_1_1_2"><span class="lab"><font color="Green" title="E18">A26</font></span></a>, <a class="txt" href="#E25:1_1_1_1_2"><span class="lab"><font color="Green" title="E20">A28</font></span></a>, <a class="txt" href="#E28:1_1_1_1_2"><span class="lab"><font color="Green" title="E23">A48</font></span></a>, <a class="txt" href="#E37:1_1_1_1_2"><span class="lab"><font color="Green" title="E28">A53</font></span></a>, <a class="txt" href="#E44:1_1_1_1_2"><span class="lab"><font color="Green" title="E31">A56</font></span></a>, <a class="txt" href="#E47:1_1_1_1_2"><span class="lab"><font color="Green" title="E33">A58</font></span></a>, <a class="txt" href="#E1:1_1_1_1_2_3_1_1"><span class="lab"><font color="Green" title="E35">A60</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_2_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_2_3_1_2"/><span class="lab"><font color="Green" title="E35">A61</font></span>: 
<font color="Maroon" title="c21">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c15">cRpL</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">j1</font> = <font color="Maroon" title="c21">j</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T20" title="NAT_1:th.20">NAT_1:20</a></span>;<br/><a NAME="E3:1_1_1_1_2_3_1_2"/>
( <font color="Maroon" title="c11">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">j1</font> is    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>) &amp; <font color="Maroon" title="c11">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">j</font> is    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>) )
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c23">pj1</font> = <font color="Maroon" title="c11">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">j1</font>, <font color="Maroon" title="c24">pj</font> = <font color="Maroon" title="c11">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">j</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T17" title="TOPMETR:th.17">TOPMETR:17</a></span>;<br/><a NAME="E5:1_1_1_1_2_3_1_2"/>
<font color="Maroon" title="c22">j1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_2_3_1_2"><span class="lab"><font color="Green" title="E35">A61</font></span></a></span>;<br/><a NAME="E6:1_1_1_1_2_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A62</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">j1</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">g</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:1_1_1_1_2"><span class="lab"><font color="Green" title="E33">A58</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E7:1_1_1_1_2_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c22">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:1_1_1_1_2"><span class="lab"><font color="Green" title="E15">A23</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E8:1_1_1_1_2_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A63</font></span>: 
<font color="Maroon" title="c23">pj1</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:1_1_1_1_2"><span class="lab"><font color="Green" title="E23">A48</font></span></a></span>;<br/><a NAME="E9:1_1_1_1_2_3_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c22">j1</font>,<font color="Maroon" title="c23">pj1</font>,<font color="Maroon" title="c24">pj</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1_1_2"><span class="lab"><font color="Green" title="E17">A25</font></span></a>, <a class="txt" href="#E25:1_1_1_1_2"><span class="lab"><font color="Green" title="E20">A28</font></span></a>, <a class="txt" href="#E28:1_1_1_1_2"><span class="lab"><font color="Green" title="E23">A48</font></span></a>, <a class="txt" href="#E44:1_1_1_1_2"><span class="lab"><font color="Green" title="E31">A56</font></span></a>, <a class="txt" href="#E6:1_1_1_1_2_3_1_2"><span class="lab"><font color="Green" title="E36">A62</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">A</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E11:1_1_1_1_2_3_1_2"/><span class="lab"><font color="Green" title="E38">A64</font></span>: 
<font color="Maroon" title="c25">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j</font>
 <span class="kw">and </span><br/><a NAME="E12:1_1_1_1_2_3_1_2"/><span class="lab"><font color="Green" title="E39">A65</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> holds  <font color="Maroon" title="c24">pj</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c23">pj1</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"><span class="p2">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist_min</a> <font color="Maroon" title="c25">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1_1_2_3_1_2"><span class="lab"><font color="Green" title="E37">A63</font></span></a></span>;<br/><a NAME="E13:1_1_1_1_2_3_1_2"/>
<font color="Maroon" title="c25">A</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1"><span class="lab"><font color="Green" title="E2">A2</font></span></a>, <a class="txt" href="#E2:1_1_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a>, <a class="txt" href="#E3:1_1_1_1_2"><span class="lab"><font color="Green" title="E8">A9</font></span></a>, <a class="txt" href="#E41:1_1_1_1_2"><span class="lab"><font color="Green" title="E30">A55</font></span></a>, <a class="txt" href="#E45:1_1_1_1_2"><span class="lab"><font color="Green" title="E32">A57</font></span></a>, <a class="txt" href="#E11:1_1_1_1_2_3_1_2"><span class="lab"><font color="Green" title="E38">A64</font></span></a>, <a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a></span>;<br/><a NAME="E14:1_1_1_1_2_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A66</font></span>: 
<font color="Maroon" title="c25">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T3" title="TOPS_1:th.3">TOPS_1:3</a></span>;<br/><a NAME="E15:1_1_1_1_2_3_1_2"/>
 not <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1_1_1_2"><span class="lab"><font color="Green" title="E29">A54</font></span></a>, <a class="txt" href="#E45:1_1_1_1_2"><span class="lab"><font color="Green" title="E32">A57</font></span></a>, <a class="txt" href="#E11:1_1_1_1_2_3_1_2"><span class="lab"><font color="Green" title="E38">A64</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E16:1_1_1_1_2_3_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist_min</a> <font color="Maroon" title="c25">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_1_1_2_3_1_2"><span class="lab"><font color="Green" title="E40">A66</font></span></a>, <a class="ref" href="hausdorf.html#T9" title="HAUSDORF:th.9">HAUSDORF:9</a></span>;<br/><a NAME="E17:1_1_1_1_2_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A67</font></span>: 
<span class="p1">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist_min</a> <font color="Maroon" title="c25">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1k.html#T9" title="JORDAN1K:th.9">JORDAN1K:9</a></span>;<br/><a NAME="E18:1_1_1_1_2_3_1_2"/>
<font color="Maroon" title="c24">pj</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c23">pj1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist_min</a> <font color="Maroon" title="c25">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:1_1_1_1_2_3_1_2"><span class="lab"><font color="Green" title="E39">A65</font></span></a></span>;<br/><a NAME="E19:1_1_1_1_2_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c24">pj</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:1_1_1_1_2_3_1_2"><span class="lab"><font color="Green" title="E41">A67</font></span></a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/><span class="kw">hence </span><a NAME="E20:1_1_1_1_2_3_1_2"/>
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c15">cRpL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:1_1_1_1_2"><span class="lab"><font color="Green" title="E15">A23</font></span></a>, <a class="txt" href="#E20:1_1_1_1_2"><span class="lab"><font color="Green" title="E18">A26</font></span></a>, <a class="txt" href="#E25:1_1_1_1_2"><span class="lab"><font color="Green" title="E20">A28</font></span></a>, <a class="txt" href="#E28:1_1_1_1_2"><span class="lab"><font color="Green" title="E23">A48</font></span></a>, <a class="txt" href="#E37:1_1_1_1_2"><span class="lab"><font color="Green" title="E28">A53</font></span></a>, <a class="txt" href="#E44:1_1_1_1_2"><span class="lab"><font color="Green" title="E31">A56</font></span></a>, <a class="txt" href="#E47:1_1_1_1_2"><span class="lab"><font color="Green" title="E33">A58</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">L</font> =  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c15">cRpL</font> as   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><span class="kw">take </span>
<font color="Maroon" title="c22">L</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>  ex <font color="Olive" title="b2">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> st <br/>( <font color="Olive" title="b2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c22">L</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">A</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c23">X</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> st <br/>( <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c23">X</font>,<font color="Maroon" title="c22">L</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> )</span><br/><span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">p</font> &amp; (  for <font color="Olive" title="b1">f1</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  st <font color="Maroon" title="c11">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f1</font> holds <br/><font color="Olive" title="b1">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">X</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c22">L</font> ) );<br/><a NAME="E51:1_1_1_1_2"/>
<font color="Maroon" title="c12">pL</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">cRpL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:1_1_1_1_2"><span class="lab"><font color="Green" title="E19">A27</font></span></a>, <a class="txt" href="#E29:1_1_1_1_2"><span class="lab"><font color="Green" title="E24">A49</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E52:1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon">S<sub>3</sub></font>[ <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">p</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_1_1_2"><span class="lab"><font color="Green" title="E10">A11</font></span></a>, <a class="txt" href="#E17:1_1_1_1_2"><span class="lab"><font color="Green" title="E15">A23</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a>, <a class="ref" href="xxreal_2.html#T3" title="XXREAL_2:th.3">XXREAL_2:3</a></span>;<br/><a NAME="E53:1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A68</font></span>: 
 ex <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>]
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c24">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E55:1_1_1_1_2"/><span class="lab"><font color="Green" title="E36">A69</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c24">k</font>]
 <span class="kw">and </span><br/><a NAME="E56:1_1_1_1_2"/><span class="lab"><font color="Green" title="E37">A70</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c24">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font>
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="#E53:1_1_1_1_2"><span class="lab"><font color="Green" title="E35">A68</font></span></a>);<br/></span><a NAME="E57:1_1_1_1_2"/><span class="lab"><font color="Green" title="E38">A71</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:1_1_1_1_2"><span class="lab"><font color="Green" title="E36">A69</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E58:1_1_1_1_2"/><span class="lab"><font color="Green" title="E39">A72</font></span>: 
<font color="Maroon" title="c24">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:1_1_1_1_2"><span class="lab"><font color="Green" title="E36">A69</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_2_4"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_1_2_4"/>
( <font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c24">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:1_1_1_1_2"><span class="lab"><font color="Green" title="E38">A71</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_2_4_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_2_4_1"/><span class="lab"><font color="Green" title="E40">A73</font></span>: 
<font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> st <br/>( <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c23">X</font>,<font color="Maroon" title="c22">L</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> )</span><br/><div class="add"><span class="kw">take </span>
<font color="Maroon" title="c8">g1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c23">X</font>,<font color="Maroon" title="c22">L</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">g1</font> )</span><br/><span class="kw">thus </span><a NAME="E2:1_1_1_1_2_4_1"/>
<font color="Maroon" title="c8">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a>, <a class="txt" href="#E3:1_1_1_1_2"><span class="lab"><font color="Green" title="E8">A9</font></span></a>, <a class="txt" href="#E9:1_1_1_1_2"><span class="lab"><font color="Green" title="E12">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c23">X</font>,<font color="Maroon" title="c22">L</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">g1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c25">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c25">y</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="c23">X</font>,<font color="Maroon" title="c22">L</font>) or <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">g1</font> )</span><br/><span class="kw">assume </span><a NAME="E3:1_1_1_1_2_4_1"/><span class="lab"><font color="Green" title="E41">A74</font></span>: 
<font color="Maroon" title="c25">y</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="c23">X</font>,<font color="Maroon" title="c22">L</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">g1</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c26">Y</font> = <font color="Maroon" title="c25">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_1_2_4_1"><span class="lab"><font color="Green" title="E41">A74</font></span></a></span>;<br/><a NAME="E5:1_1_1_1_2_4_1"/><span class="lab"><font color="Green" title="E42">A75</font></span>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c23">X</font>,<font color="Maroon" title="c26">Y</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_1_2_4_1"><span class="lab"><font color="Green" title="E41">A74</font></span></a>, <a class="ref" href="metric_1.html#T11" title="METRIC_1:th.11">METRIC_1:11</a></span>;<br/><span class="kw">assume </span><a NAME="E6:1_1_1_1_2_4_1"/>
 not <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">g1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E7:1_1_1_1_2_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c26">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">g19</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E8:1_1_1_1_2_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A76</font></span>: 
<span class="p1">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist_min</a> <font color="Maroon" title="c9">g19</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">X</font> <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="c23">X</font>,<font color="Maroon" title="c26">Y</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hausdorf.html#T13" title="HAUSDORF:th.13">HAUSDORF:13</a></span>;<br/><a NAME="E9:1_1_1_1_2_4_1"/>
<span class="p1">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist_min</a> <font color="Maroon" title="c9">g19</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">X</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c22">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_1_1_2"><span class="lab"><font color="Green" title="E9">A10</font></span></a>, <a class="txt" href="#E18:1_1_1_1_2"><span class="lab"><font color="Green" title="E16">A24</font></span></a>, <a class="txt" href="#E55:1_1_1_1_2"><span class="lab"><font color="Green" title="E36">A69</font></span></a>, <a class="txt" href="#E1:1_1_1_1_2_4_1"><span class="lab"><font color="Green" title="E40">A73</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E10:1_1_1_1_2_4_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1_1_2_4_1"><span class="lab"><font color="Green" title="E42">A75</font></span></a>, <a class="txt" href="#E8:1_1_1_1_2_4_1"><span class="lab"><font color="Green" title="E43">A76</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_2_4_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_2_4_2"/><span class="lab"><font color="Green" title="E40">A77</font></span>: 
<font color="Maroon" title="c24">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> st <br/>( <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c23">X</font>,<font color="Maroon" title="c22">L</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c25">k1</font> = <font color="Maroon" title="c24">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T20" title="NAT_1:th.20">NAT_1:20</a></span>;<br/><a NAME="E3:1_1_1_1_2_4_2"/>
( <font color="Maroon" title="c11">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">k1</font> is    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>) &amp; <font color="Maroon" title="c11">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">k</font> is    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>) )
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c26">pk1</font> = <font color="Maroon" title="c11">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">k1</font>, <font color="Maroon" title="c27">pk</font> = <font color="Maroon" title="c11">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">k</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T17" title="TOPMETR:th.17">TOPMETR:17</a></span>;<br/><a NAME="E5:1_1_1_1_2_4_2"/>
<font color="Maroon" title="c25">k1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_2_4_2"><span class="lab"><font color="Green" title="E40">A77</font></span></a></span>;<br/><a NAME="E6:1_1_1_1_2_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A78</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">k1</font> &amp; <font color="Maroon" title="c25">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">g</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:1_1_1_1_2"><span class="lab"><font color="Green" title="E15">A23</font></span></a>, <a class="txt" href="#E58:1_1_1_1_2"><span class="lab"><font color="Green" title="E39">A72</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E7:1_1_1_1_2_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c25">k1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:1_1_1_1_2"><span class="lab"><font color="Green" title="E15">A23</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E8:1_1_1_1_2_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A79</font></span>: 
<font color="Maroon" title="c26">pk1</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:1_1_1_1_2"><span class="lab"><font color="Green" title="E23">A48</font></span></a></span>;<br/><a NAME="E9:1_1_1_1_2_4_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c25">k1</font>,<font color="Maroon" title="c26">pk1</font>,<font color="Maroon" title="c27">pk</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1_1_2"><span class="lab"><font color="Green" title="E17">A25</font></span></a>, <a class="txt" href="#E28:1_1_1_1_2"><span class="lab"><font color="Green" title="E23">A48</font></span></a>, <a class="txt" href="#E55:1_1_1_1_2"><span class="lab"><font color="Green" title="E36">A69</font></span></a>, <a class="txt" href="#E6:1_1_1_1_2_4_2"><span class="lab"><font color="Green" title="E41">A78</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c28">A</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E11:1_1_1_1_2_4_2"/><span class="lab"><font color="Green" title="E43">A80</font></span>: 
<font color="Maroon" title="c28">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font>
 <span class="kw">and </span><br/><a NAME="E12:1_1_1_1_2_4_2"/><span class="lab"><font color="Green" title="E44">A81</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> holds  <font color="Maroon" title="c27">pk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c26">pk1</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"><span class="p2">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist_min</a> <font color="Maroon" title="c28">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1_1_2_4_2"><span class="lab"><font color="Green" title="E42">A79</font></span></a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c28">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c28">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c23">X</font>,<font color="Maroon" title="c22">L</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c28">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )</span><br/><a NAME="E13:1_1_1_1_2_4_2"/>
<font color="Maroon" title="c28">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_1_2"><span class="lab"><font color="Green" title="E8">A9</font></span></a>, <a class="txt" href="#E25:1_1_1_1_2"><span class="lab"><font color="Green" title="E20">A28</font></span></a>, <a class="txt" href="#E55:1_1_1_1_2"><span class="lab"><font color="Green" title="E36">A69</font></span></a>, <a class="txt" href="#E11:1_1_1_1_2_4_2"><span class="lab"><font color="Green" title="E43">A80</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E14:1_1_1_1_2_4_2"/>
<font color="Maroon" title="c28">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c23">X</font>,<font color="Maroon" title="c22">L</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c28">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c29">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c29">y</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="c23">X</font>,<font color="Maroon" title="c22">L</font>) or <font color="Maroon" title="c29">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )</span><br/><span class="kw">assume </span><a NAME="E15:1_1_1_1_2_4_2"/><span class="lab"><font color="Green" title="E45">A82</font></span>: 
<font color="Maroon" title="c29">y</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="c23">X</font>,<font color="Maroon" title="c22">L</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c29">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span><br/><span class="kw">reconsider </span><font color="Maroon" title="c30">Y</font> = <font color="Maroon" title="c29">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:1_1_1_1_2_4_2"><span class="lab"><font color="Green" title="E45">A82</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E17:1_1_1_1_2_4_2"/>
 not <font color="Maroon" title="c29">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E18:1_1_1_1_2_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c30">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E19:1_1_1_1_2_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A83</font></span>: 
<span class="p1">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist_min</a> <font color="Maroon" title="c28">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">X</font> <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="c23">X</font>,<font color="Maroon" title="c30">Y</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hausdorf.html#T13" title="HAUSDORF:th.13">HAUSDORF:13</a></span>;<br/><a NAME="E20:1_1_1_1_2_4_2"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c23">X</font>,<font color="Maroon" title="c30">Y</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:1_1_1_1_2_4_2"><span class="lab"><font color="Green" title="E45">A82</font></span></a>, <a class="ref" href="metric_1.html#T11" title="METRIC_1:th.11">METRIC_1:11</a></span>;<br/><a NAME="E21:1_1_1_1_2_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A84</font></span>: 
<span class="p1">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist_min</a> <font color="Maroon" title="c28">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">X</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1_1_2_4_2"><span class="lab"><font color="Green" title="E46">A83</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E22:1_1_1_1_2_4_2"/><span class="lab"><font color="Green" title="E48">A85</font></span>: 
<font color="Maroon" title="c27">pk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">X</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c22">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:1_1_1_1_2"><span class="lab"><font color="Green" title="E36">A69</font></span></a></span>;<br/><a NAME="E23:1_1_1_1_2_4_2"/>
<font color="Maroon" title="c27">pk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c26">pk1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">X</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist_min</a> <font color="Maroon" title="c28">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">X</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:1_1_1_1_2_4_2"><span class="lab"><font color="Green" title="E44">A81</font></span></a></span>;<br/><a NAME="E24:1_1_1_1_2_4_2"/><span class="kw">then </span>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c25">k1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:1_1_1_1_2"><span class="lab"><font color="Green" title="E15">A23</font></span></a>, <a class="txt" href="#E6:1_1_1_1_2_4_2"><span class="lab"><font color="Green" title="E41">A78</font></span></a>, <a class="txt" href="#E21:1_1_1_1_2_4_2"><span class="lab"><font color="Green" title="E47">A84</font></span></a>, <a class="txt" href="#E22:1_1_1_1_2_4_2"><span class="lab"><font color="Green" title="E48">A85</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a>, <a class="ref" href="xxreal_0.html#T16" title="XXREAL_0:th.16">XXREAL_0:16</a></span>;<br/><a NAME="E25:1_1_1_1_2_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c25">k1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c25">k1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:1_1_1_1_2"><span class="lab"><font color="Green" title="E37">A70</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E26:1_1_1_1_2_4_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
