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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">M</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</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="c1">M</font> is  <a href="tbsp_1.html#V4" title="TBSP_1:attr.4">complete</a>  implies  for <font color="Olive" title="b1">A</font> being   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 <font color="Maroon" title="c1">M</font><br/>  for <font color="Olive" title="b2">A9</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 <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">A9</font> holds <br/>( <font color="Maroon" title="c1">M</font> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Olive" title="b1">A</font> is  <a href="tbsp_1.html#V4" title="TBSP_1:attr.4">complete</a>  iff <font color="Olive" title="b2">A9</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:32"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
<font color="Maroon" title="c1">M</font> is  <a href="tbsp_1.html#V4" title="TBSP_1:attr.4">complete</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <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 <font color="Maroon" title="c1">M</font><br/>  for <font color="Olive" title="b2">A9</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 <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">A9</font> holds <br/>( <font color="Maroon" title="c1">M</font> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Olive" title="b1">A</font> is  <a href="tbsp_1.html#V4" title="TBSP_1:attr.4">complete</a>  iff <font color="Olive" title="b2">A9</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">T</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">let </span><font color="Maroon" title="c3">A</font> be   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 <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">  for <font color="Olive" title="b1">A9</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 <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A9</font> holds <br/>( <font color="Maroon" title="c1">M</font> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c3">A</font> is  <a href="tbsp_1.html#V4" title="TBSP_1:attr.4">complete</a>  iff <font color="Olive" title="b1">A9</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">A9</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#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="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A9</font> implies ( <font color="Maroon" title="c1">M</font> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c3">A</font> is  <a href="tbsp_1.html#V4" title="TBSP_1:attr.4">complete</a>  iff <font color="Maroon" title="c4">A9</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  ) )</span><br/>



<span class="kw">assume </span><a NAME="E2:32"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">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="c1">M</font> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c3">A</font> is  <a href="tbsp_1.html#V4" title="TBSP_1:attr.4">complete</a>  iff <font color="Maroon" title="c4">A9</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">MA</font> = <font color="Maroon" title="c1">M</font> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c3">A</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">TA</font> =  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c3">A</font></span>)</span>;<br/>
<span class="kw">thus </span><a NAME="E3:32"/>
( <font color="Maroon" title="c1">M</font> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c3">A</font> is  <a href="tbsp_1.html#V4" title="TBSP_1:attr.4">complete</a>  implies <font color="Maroon" title="c4">A9</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</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="c4">A9</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  implies <font color="Maroon" title="c1">M</font> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c3">A</font> is  <a href="tbsp_1.html#V4" title="TBSP_1:attr.4">complete</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:32_1"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
<font color="Maroon" title="c1">M</font> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c3">A</font> is  <a href="tbsp_1.html#V4" title="TBSP_1:attr.4">complete</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">A9</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> </span><br/>

<a NAME="E2:32_1"/><span class="lab"><font color="Green" title="E29">A4</font></span>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c4">A9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">p</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="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c4">A9</font> or <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A9</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_1_1"/><span class="lab"><font color="Green" title="E30">A5</font></span>: 
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c4">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="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A9</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">p</font> = <font color="Maroon" title="c7">p</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="#E1:32_1_1"><span class="lab"><font color="Green" title="E30">A5</font></span></a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>)</span>)</span>;<br/>
<a NAME="E3:32_1_1"/><span class="lab"><font color="Green" title="E31">A6</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ex <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c3">A</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  implies  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c3">A</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_1_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c3">A</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">i</font> = <font color="Maroon" title="c9">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</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="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c3">A</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span>] )</span><br/>

<a NAME="E3:32_1_1_1"/>
<font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E4:32_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c3">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#D2" title="TOPMETR:def.2">TOPMETR:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:32_1_1_1"/>
( <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c3">A</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</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>
<span class="kw">consider </span><font color="Maroon" title="c9">f</font> being   <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c3">A</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E5:32_1_1"/><span class="lab"><font color="Green" title="E32">A7</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="#E3:32_1_1"><span class="lab"><font color="Green" title="E31">A6</font></span></a>);<br/></span>
<div><span class="lab"><font color="Green" title="E33">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">f</font> implies  not <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:32_1_1_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">f</font>
 ; <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="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> </span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">i</font> = <font color="Maroon" title="c10">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">B</font> =  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="E4:32_1_1_2"/>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">Family_open_set</a> <font color="Maroon" title="c1">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pcomps_1.html#T29" title="PCOMPS_1:th.29">PCOMPS_1:29</a></span>;<br/><a NAME="E5:32_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A9</font></span>: 
<font color="Maroon" title="c12">B</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="ref" href="pre_topc.html#D2" title="PRE_TOPC:def.2">PRE_TOPC:def 2</a></span>;<br/><a NAME="E6:32_1_1_2"/>
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tbsp_1.html#T11" title="TBSP_1:th.11">TBSP_1:11</a>, <a class="ref" href="xreal_1.html#T139" title="XREAL_1:th.139">XREAL_1:139</a></span>;<br/><a NAME="E7:32_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">B</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">A9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="txt" href="#E5:32_1_1_2"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="ref" href="pre_topc.html#T24" title="PRE_TOPC:th.24">PRE_TOPC:24</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E9:32_1_1_2"/><span class="lab"><font color="Green" title="E35">A10</font></span>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">B</font>
 <span class="kw">and </span><br/><a NAME="E10:32_1_1_2"/><span class="lab"><font color="Green" title="E36">A11</font></span>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">y</font> = <font color="Maroon" title="c13">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="#E9:32_1_1_2"><span class="lab"><font color="Green" title="E35">A10</font></span></a></span>;<br/><a NAME="E12:32_1_1_2"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c14">y</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32_1_1_2"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="ref" href="metric_1.html#T11" title="METRIC_1:th.11">METRIC_1:11</a></span>;<br/><a NAME="E13:32_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></span>;<br/><a NAME="E14:32_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E10:32_1_1_2"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E15:32_1_1_2"/>
 not <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</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="#E5:32_1_1"><span class="lab"><font color="Green" title="E32">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><span class="lab"><font color="Green" title="E34">A12</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">i</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">i</font> is  <a href="compl_sp.html#V3" title="COMPL_SP:attr.3">closed</a> </span><br/><span class="kw">reconsider </span><font color="Maroon" title="c11">cB</font> =  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></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/><span class="kw">reconsider </span><font color="Maroon" title="c12">fi</font> = <font color="Maroon" title="c9">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">i</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> ;<br/><a NAME="E3:32_1_1_3"/><span class="lab"><font color="Green" title="E35">A13</font></span>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E4:32_1_1_3"/>
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c11">cB</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">Family_open_set</a> <font color="Maroon" title="c1">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nagata_1.html#T14" title="NAGATA_1:th.14">NAGATA_1:14</a></span>;<br/><a NAME="E5:32_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c11">cB</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="ref" href="pre_topc.html#D2" title="PRE_TOPC:def.2">PRE_TOPC:def 2</a></span>;<br/><a NAME="E6:32_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A14</font></span>: 
<font color="Maroon" title="c11">cB</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="E7:32_1_1_3"/><span class="lab"><font color="Green" title="E37">A15</font></span>: 
 <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">A9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="hausdorf.html#T16" title="HAUSDORF:th.16">HAUSDORF:16</a></span>;<br/><a NAME="E8:32_1_1_3"/><span class="kw">then </span>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><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> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">A9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#D2" title="TOPMETR:def.2">TOPMETR:def 2</a></span>;<br/><a NAME="E9:32_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c12">fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">cB</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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"><span class="p3">(<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="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">A9</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_1_1"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="txt" href="#E3:32_1_1_3"><span class="lab"><font color="Green" title="E35">A13</font></span></a></span>;<br/><a NAME="E10:32_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c12">fi</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="txt" href="#E6:32_1_1_3"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="txt" href="#E7:32_1_1_3"><span class="lab"><font color="Green" title="E37">A15</font></span></a>, <a class="ref" href="pre_topc.html#T13" title="PRE_TOPC:th.13">PRE_TOPC:13</a></span>;<br/><span class="kw">hence </span><a NAME="E11:32_1_1_3"/>
<font color="Maroon" title="c9">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">i</font> is  <a href="compl_sp.html#V3" title="COMPL_SP:attr.3">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="compl_sp.html#T6" target="_self" title="COMPL_SP:th.6">Th6</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="hs2(this)" href="javascript:()" title="32_1_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">i</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">i</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> </span><br/><span class="kw">set </span><font color="Maroon" title="c11">ACL</font> = <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>)</span>)</span>;<br/><a NAME="E1:32_1_1_4"/>
 <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>) is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal6.html#T59" title="TOPREAL6:th.59">TOPREAL6:59</a></span>;<br/><a NAME="E2:32_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A16</font></span>: 
<font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>)</span>)</span> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tbsp_1.html#T14" title="TBSP_1:th.14">TBSP_1:14</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/><a NAME="E3:32_1_1_4"/>
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E4:32_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c9">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_1_1"><span class="lab"><font color="Green" title="E32">A7</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:32_1_1_4"/>
<font color="Maroon" title="c9">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">i</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_1_1_4"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="ref" href="compl_sp.html#T15" target="_self" title="COMPL_SP:th.15">Th15</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>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">f</font> = <font color="Maroon" title="c9">f</font> as   <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>   <a href="compl_sp.html#V1" title="COMPL_SP:attr.1">bounded</a>   <a href="compl_sp.html#V7" title="COMPL_SP:attr.7">closed</a>  <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c3">A</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1_1"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="txt" href="#E7:32_1_1"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="ref" href="compl_sp.html#D1" target="_self" title="COMPL_SP:def.1">Def1</a>, <a class="ref" href="compl_sp.html#D8" target="_self" title="COMPL_SP:def.8">Def8</a>, <a class="ref" href="funct_1.html#D9" title="FUNCT_1:def.9">FUNCT_1:def 9</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">df</font> =  <a href="compl_sp.html#K1" title="COMPL_SP:func.1">diameter</a> <font color="Maroon" title="c10">f</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">NULL</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <font color="Maroon" title="c13">TWO</font> = 2 as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  = 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">seq</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><span class="kw"> such that </span><br/><a NAME="E12:32_1_1"/><span class="lab"><font color="Green" title="E35">A17</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c14">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="seq_1.html#S1" title="SEQ_1:sch.1">SEQ_1:sch 1</a>();<br/></span>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <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="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">i</font></span><br/><span class="kw">set </span><font color="Maroon" title="c16">i1</font> = <font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/><a NAME="E1:32_1_1_5"/>
 <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">x</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="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>) or <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_1_1_5_1"/><span class="lab"><font color="Green" title="E36">A18</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c18">q</font> = <font color="Maroon" title="c17">x</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="#E1:32_1_1_5_1"><span class="lab"><font color="Green" title="E36">A18</font></span></a></span>;<br/>
<a NAME="E3:32_1_1_5_1"/>
<font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<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>;<br/>
<a NAME="E4:32_1_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A19</font></span>: 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T76" title="XREAL_1:th.76">XREAL_1:76</a></span>;<br/>
<a NAME="E5:32_1_1_5_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c18">q</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_5_1"><span class="lab"><font color="Green" title="E36">A18</font></span></a>, <a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></span>;<br/>
<a NAME="E6:32_1_1_5_1"/><span class="kw">then </span>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c18">q</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_1_5_1"><span class="lab"><font color="Green" title="E37">A19</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:32_1_1_5_1"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</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><a NAME="E2:32_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A20</font></span>: 
<font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></span>;<br/><a NAME="E3:32_1_1_5"/>
<font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_1_1"><span class="lab"><font color="Green" title="E32">A7</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:32_1_1_5"/>
<font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_1_1"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="txt" href="#E2:32_1_1_5"><span class="lab"><font color="Green" title="E36">A20</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>
<a NAME="E14:32_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A21</font></span>: 
<font color="Maroon" title="c10">f</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="kurato_0.html#D3" title="KURATO_0:def.3">KURATO_0:def 3</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">Ts</font> = <font color="Maroon" title="c13">TWO</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c14">seq</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">Ns</font> = <font color="Maroon" title="c12">NULL</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c14">seq</font>;<br/>
<a NAME="E15:32_1_1"/><span class="lab"><font color="Green" title="E37">A22</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c14">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:32_1_1"><span class="lab"><font color="Green" title="E35">A17</font></span></a></span>;<br/>
<a NAME="E16:32_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A23</font></span>: 
<font color="Maroon" title="c12">NULL</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c14">seq</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_2.html#T7" title="SEQ_2:th.7">SEQ_2:7</a>, <a class="ref" href="seq_4.html#T30" title="SEQ_4:th.30">SEQ_4:30</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E39">A24</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c12">NULL</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c14">seq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K1" title="COMPL_SP:func.1">diameter</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">n</font> &amp; <span class="p1">(<span class="default"><a href="compl_sp.html#K1" title="COMPL_SP:func.1">diameter</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">TWO</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c14">seq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">n</font> )</span><br/><span class="kw">set </span><font color="Maroon" title="c18">cB</font> =  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>);<br/><a NAME="E1:32_1_1_6"/><span class="lab"><font color="Green" title="E40">A25</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">NULL</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c14">seq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">NULL</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T9" title="SEQ_1:th.9">SEQ_1:9</a></span>;<br/><a NAME="E2:32_1_1_6"/><span class="lab"><font color="Green" title="E41">A26</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">TWO</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c14">seq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">TWO</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T9" title="SEQ_1:th.9">SEQ_1:9</a></span>;<br/><a NAME="E3:32_1_1_6"/><span class="lab"><font color="Green" title="E42">A27</font></span>: 
 <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>) is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal6.html#T59" title="TOPREAL6:th.59">TOPREAL6:59</a></span>;<br/><a NAME="E4:32_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A28</font></span>: 
<font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tbsp_1.html#T14" title="TBSP_1:th.14">TBSP_1:14</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/><a NAME="E5:32_1_1_6"/><span class="lab"><font color="Green" title="E44">A29</font></span>: 
 <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_1_6"><span class="lab"><font color="Green" title="E42">A27</font></span></a>, <a class="ref" href="tbsp_1.html#T24" title="TBSP_1:th.24">TBSP_1:24</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/><a NAME="E6:32_1_1_6"/>
 <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="compl_sp.html#T5" target="_self" title="COMPL_SP:th.5">Th5</a></span>;<br/><a NAME="E7:32_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A30</font></span>: 
 <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_1_1_6"><span class="lab"><font color="Green" title="E44">A29</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E8:32_1_1_6"/><span class="lab"><font color="Green" title="E46">A31</font></span>: 
<font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_1_1"><span class="lab"><font color="Green" title="E32">A7</font></span></a></span>;<br/><a NAME="E9:32_1_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">n</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_1_6"><span class="lab"><font color="Green" title="E43">A28</font></span></a>, <a class="ref" href="compl_sp.html#T15" target="_self" title="COMPL_SP:th.15">Th15</a></span>;<br/><a NAME="E10:32_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A32</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tbsp_1.html#T21" title="TBSP_1:th.21">TBSP_1:21</a></span>;<br/><a NAME="E11:32_1_1_6"/>
 <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_1_6"><span class="lab"><font color="Green" title="E43">A28</font></span></a>, <a class="txt" href="#E8:32_1_1_6"><span class="lab"><font color="Green" title="E46">A31</font></span></a>, <a class="ref" href="compl_sp.html#T16" target="_self" title="COMPL_SP:th.16">Th16</a></span>;<br/><a NAME="E12:32_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A33</font></span>: 
 <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:32_1_1_6"><span class="lab"><font color="Green" title="E45">A30</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E13:32_1_1_6"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:32_1_1"><span class="lab"><font color="Green" title="E35">A17</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E14:32_1_1_6"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c12">NULL</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c14">seq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K1" title="COMPL_SP:func.1">diameter</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">n</font> &amp; <span class="p1">(<span class="default"><a href="compl_sp.html#K1" title="COMPL_SP:func.1">diameter</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">TWO</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c14">seq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:32_1_1_6"><span class="lab"><font color="Green" title="E47">A32</font></span></a>, <a class="txt" href="#E12:32_1_1_6"><span class="lab"><font color="Green" title="E48">A33</font></span></a>, <a class="txt" href="#E1:32_1_1_6"><span class="lab"><font color="Green" title="E40">A25</font></span></a>, <a class="txt" href="#E2:32_1_1_6"><span class="lab"><font color="Green" title="E41">A26</font></span></a>, <a class="ref" href="compl_sp.html#D2" target="_self" title="COMPL_SP:def.2">Def2</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>
<a NAME="E18:32_1_1"/><span class="lab"><font color="Green" title="E40">A34</font></span>: 
<font color="Maroon" title="c13">TWO</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c14">seq</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:32_1_1"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="ref" href="seq_2.html#T7" title="SEQ_2:th.7">SEQ_2:7</a>, <a class="ref" href="seq_4.html#T30" title="SEQ_4:th.30">SEQ_4:30</a></span>;<br/>
<a NAME="E19:32_1_1"/><span class="lab"><font color="Green" title="E41">A35</font></span>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c14">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:32_1_1"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="ref" href="seq_4.html#T30" title="SEQ_4:th.30">SEQ_4:30</a></span>;<br/>
<a NAME="E20:32_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A36</font></span>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">TWO</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c14">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">TWO</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:32_1_1"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="ref" href="seq_2.html#T8" title="SEQ_2:th.8">SEQ_2:8</a>, <a class="ref" href="seq_4.html#T30" title="SEQ_4:th.30">SEQ_4:30</a></span>;<br/>
<a NAME="E21:32_1_1"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">NULL</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c14">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">NULL</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:32_1_1"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="txt" href="#E19:32_1_1"><span class="lab"><font color="Green" title="E41">A35</font></span></a>, <a class="ref" href="seq_2.html#T8" title="SEQ_2:th.8">SEQ_2:8</a>, <a class="ref" href="seq_4.html#T30" title="SEQ_4:th.30">SEQ_4:30</a></span>;<br/>
<a NAME="E22:32_1_1"/><span class="kw">then </span>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K1" title="COMPL_SP:func.1">diameter</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:32_1_1"><span class="lab"><font color="Green" title="E38">A23</font></span></a>, <a class="txt" href="#E18:32_1_1"><span class="lab"><font color="Green" title="E40">A34</font></span></a>, <a class="txt" href="#E20:32_1_1"><span class="lab"><font color="Green" title="E42">A36</font></span></a>, <a class="txt" href="#E17:32_1_1"><span class="lab"><font color="Green" title="E39">A24</font></span></a>, <a class="ref" href="seq_2.html#T20" title="SEQ_2:th.20">SEQ_2:20</a></span>;<br/>
<a NAME="E23:32_1_1"/><span class="kw">then </span>
 not  <a href="kurato_0.html#K2" title="KURATO_0:func.2">meet</a> <font color="Maroon" title="c10">f</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="#E1:32_1"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E14:32_1_1"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="ref" href="compl_sp.html#T10" target="_self" title="COMPL_SP:th.10">Th10</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">q</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E25:32_1_1"/><span class="lab"><font color="Green" title="E43">A37</font></span>: 
<font color="Maroon" title="c17">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_0.html#K2" title="KURATO_0:func.2">meet</a> <font color="Maroon" title="c10">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">q</font> = <font color="Maroon" title="c17">q</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="#E25:32_1_1"><span class="lab"><font color="Green" title="E43">A37</font></span></a>, <a class="ref" href="topmetr.html#T8" title="TOPMETR:th.8">TOPMETR:8</a></span>;<br/>
<a NAME="E27:32_1_1"/><span class="lab"><font color="Green" title="E44">A38</font></span>: 
<font color="Maroon" title="c14">seq</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:32_1_1"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="ref" href="seq_4.html#T30" title="SEQ_4:th.30">SEQ_4:30</a></span>;<br/>
<a NAME="E28:32_1_1"/>
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:32_1_1_7"/>
<font color="Maroon" title="c8">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">q</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="E2:32_1_1_7"/><span class="kw">then </span>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c18">q</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T2" title="METRIC_1:th.2">METRIC_1:2</a></span>;<br/>
<a NAME="E3:32_1_1_7"/><span class="kw">then </span>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c18">q</font>) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T5" title="METRIC_1:th.5">METRIC_1:5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E5:32_1_1_7"/><span class="lab"><font color="Green" title="E45">A39</font></span>: 
 for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c19">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c18">q</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:32_1_1"><span class="lab"><font color="Green" title="E44">A38</font></span></a>, <a class="txt" href="#E19:32_1_1"><span class="lab"><font color="Green" title="E41">A35</font></span></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c20">cB</font> =  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>);<br/>
<a NAME="E6:32_1_1_7"/><span class="lab"><font color="Green" title="E46">A40</font></span>: 
<font color="Maroon" title="c18">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:32_1_1"><span class="lab"><font color="Green" title="E43">A37</font></span></a>, <a class="ref" href="kurato_0.html#T3" title="KURATO_0:th.3">KURATO_0:3</a></span>;<br/>
<a NAME="E7:32_1_1_7"/>
<font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_1_1"><span class="lab"><font color="Green" title="E32">A7</font></span></a></span>;<br/>
<a NAME="E8:32_1_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c18">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1_1_7"><span class="lab"><font color="Green" title="E46">A40</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E9:32_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A41</font></span>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c18">q</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c19">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></span>;<br/>
<a NAME="E10:32_1_1_7"/>
<font color="Maroon" title="c14">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c19">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:32_1_1"><span class="lab"><font color="Green" title="E35">A17</font></span></a></span>;<br/>
<a NAME="E11:32_1_1_7"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c19">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:32_1_1_7"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_1_1_7"><span class="lab"><font color="Green" title="E45">A39</font></span></a>, <a class="txt" href="#E9:32_1_1_7"><span class="lab"><font color="Green" title="E47">A41</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>
<a NAME="E29:32_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A42</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:32_1_1"><span class="lab"><font color="Green" title="E43">A37</font></span></a>, <a class="ref" href="kurato_0.html#T3" title="KURATO_0:th.3">KURATO_0:3</a></span>;<br/>
<a NAME="E30:32_1_1"/>
<font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c8">p</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_1_1"><span class="lab"><font color="Green" title="E32">A7</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E31:32_1_1"/>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E29:32_1_1"><span class="lab"><font color="Green" title="E45">A42</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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>
<a NAME="E3:32_1"/>
<font color="Maroon" title="c4">A9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c4">A9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T18" title="PRE_TOPC:th.18">PRE_TOPC:18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:32_1"/>
<font color="Maroon" title="c4">A9</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="txt" href="#E2:32_1"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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>

<span class="kw">assume </span><a NAME="E4:32"/><span class="lab"><font color="Green" title="E28">A43</font></span>: 
<font color="Maroon" title="c4">A9</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</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="c1">M</font> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c3">A</font> is  <a href="tbsp_1.html#V4" title="TBSP_1:attr.4">complete</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">S</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c3">A</font></span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tbsp_1.html#D5" title="TBSP_1:def.5">TBSP_1:def 5</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="c7">S</font> is  <a href="tbsp_1.html#V3" title="TBSP_1:attr.3">Cauchy</a>  or <font color="Maroon" title="c7">S</font> is  <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E5:32"/><span class="lab"><font color="Green" title="E29">A44</font></span>: 
<font color="Maroon" title="c7">S</font> is  <a href="tbsp_1.html#V3" title="TBSP_1:attr.3">Cauchy</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="c7">S</font> is  <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">S9</font> = <font color="Maroon" title="c7">S</font> as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">M</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="compl_sp.html#T17" target="_self" title="COMPL_SP:th.17">Th17</a></span>;<br/>
<a NAME="E7:32"/>
<font color="Maroon" title="c8">S9</font> is  <a href="tbsp_1.html#V3" title="TBSP_1:attr.3">Cauchy</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32"><span class="lab"><font color="Green" title="E29">A44</font></span></a>, <a class="ref" href="compl_sp.html#T18" target="_self" title="COMPL_SP:th.18">Th18</a></span>;<br/>
<a NAME="E8:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A45</font></span>: 
<font color="Maroon" title="c8">S9</font> is  <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="tbsp_1.html#D5" title="TBSP_1:def.5">TBSP_1:def 5</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E31">A46</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <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">S9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A9</font></span><br/><a NAME="E1:32_2"/>
<font color="Maroon" title="c7">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c3">A</font></span>)</span>
 ;<br/><span class="kw">hence </span><a NAME="E2:32_2"/>
<font color="Maroon" title="c8">S9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="topmetr.html#D2" title="TOPMETR:def.2">TOPMETR:def 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>
<a NAME="E10:32"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="topmetr.html#D2" title="TOPMETR:def.2">TOPMETR:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">limS</font> =  <a href="tbsp_1.html#K1" title="TBSP_1:func.1">lim</a> <font color="Maroon" title="c8">S9</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c3">A</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32"><span class="lab"><font color="Green" title="E28">A43</font></span></a>, <a class="txt" href="#E8:32"><span class="lab"><font color="Green" title="E30">A45</font></span></a>, <a class="txt" href="#E9:32"><span class="lab"><font color="Green" title="E31">A46</font></span></a>, <a class="ref" href="topmetr3.html#T1" title="TOPMETR3:th.1">TOPMETR3:1</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c9">limS</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tbsp_1.html#D2" title="TBSP_1:def.2">TBSP_1:def 2</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">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  ex <font color="Olive">b<sub>2</sub></font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/> for <font color="Olive">b<sub>3</sub></font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span>,<font color="Maroon" title="c9">limS</font>) ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c10">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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="c10">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  ex <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/> for <font color="Olive">b<sub>2</sub></font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> or  not <font color="Maroon" title="c10">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<font color="Maroon" title="c9">limS</font>) ) )</span><br/>

<span class="kw">assume </span><a NAME="E12:32"/>
<font color="Maroon" title="c10">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/> for <font color="Olive">b<sub>2</sub></font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> or  not <font color="Maroon" title="c10">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<font color="Maroon" title="c9">limS</font>) )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E14:32"/><span class="lab"><font color="Green" title="E32">A47</font></span>: 
 for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">n</font> holds <br/> <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">S9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="tbsp_1.html#K1" title="TBSP_1:func.1">lim</a> <font color="Maroon" title="c8">S9</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32"><span class="lab"><font color="Green" title="E30">A45</font></span></a>, <a class="ref" href="tbsp_1.html#D3" title="TBSP_1:def.3">TBSP_1:def 3</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c11">n</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">b<sub>1</sub></font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <br/>(  not <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Maroon" title="c10">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Maroon" title="c9">limS</font>) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c12">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <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="c11">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> or  not <font color="Maroon" title="c10">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span>,<font color="Maroon" title="c9">limS</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E15:32"/><span class="lab"><font color="Green" title="E33">A48</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">n</font>
 ; <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="c10">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span>,<font color="Maroon" title="c9">limS</font>)</span><br/>

<a NAME="E16:32"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">m</font></span>)</span>,<font color="Maroon" title="c9">limS</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">S9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="tbsp_1.html#K1" title="TBSP_1:func.1">lim</a> <font color="Maroon" title="c8">S9</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#D1" title="TOPMETR:def.1">TOPMETR:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:32"/>
 not <font color="Maroon" title="c10">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span>,<font color="Maroon" title="c9">limS</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:32"><span class="lab"><font color="Green" title="E32">A47</font></span></a>, <a class="txt" href="#E15:32"><span class="lab"><font color="Green" title="E33">A48</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>
