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<span class="kw">let </span><font color="Maroon" title="c1">TM</font> be   <a href="pcomps_1.html#V3" title="PCOMPS_1:attr.3">metrizable</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Null</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font>  st <font color="Maroon" title="c1">TM</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">Null</font> is  <a href="waybel23.html#V5" title="WAYBEL23:attr.5">second-countable</a>  holds <br/>( ( <font color="Olive" title="b1">Null</font> is  <a href="topdim_1.html#NV2" title="TOPDIM_1:NV.2">finite-ind</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Olive" title="b1">Null</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) iff  ex <font color="Olive" title="b2">B</font> being   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c1">TM</font> st <br/> for <font color="Olive" title="b3">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font>  st <font color="Olive" title="b3">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">B</font> holds <br/><font color="Olive" title="b1">Null</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b3">A</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">cTM</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">TM</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">TOP</font> =  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">TM</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">Null</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">TM</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">Null</font> is  <a href="waybel23.html#V5" title="WAYBEL23:attr.5">second-countable</a>  implies ( ( <font color="Maroon" title="c4">Null</font> is  <a href="topdim_1.html#NV2" title="TOPDIM_1:NV.2">finite-ind</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Maroon" title="c4">Null</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) iff  ex <font color="Olive" title="b1">B</font> being   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c1">TM</font> st <br/> for <font color="Olive" title="b2">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font>  st <font color="Olive" title="b2">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Maroon" title="c4">Null</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b2">A</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:63"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
<font color="Maroon" title="c1">TM</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">Null</font> is  <a href="waybel23.html#V5" title="WAYBEL23:attr.5">second-countable</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c4">Null</font> is  <a href="topdim_1.html#NV2" title="TOPDIM_1:NV.2">finite-ind</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Maroon" title="c4">Null</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) iff  ex <font color="Olive" title="b1">B</font> being   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c1">TM</font> st <br/> for <font color="Olive" title="b2">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font>  st <font color="Olive" title="b2">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Maroon" title="c4">Null</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b2">A</font> )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">B</font> being   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c1">TM</font> st <br/> for <font color="Olive" title="b2">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font>  st <font color="Olive" title="b2">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Maroon" title="c4">Null</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b2">A</font> implies ( <font color="Maroon" title="c4">Null</font> is  <a href="topdim_1.html#NV2" title="TOPDIM_1:NV.2">finite-ind</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Maroon" title="c4">Null</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) )</span>
<div class="add"><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">TM</font><br/>  for <font color="Olive" title="b2">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">TM</font> &amp; (  not <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">A</font> implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">TM</font> ) &amp; ( <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">A</font> implies  ex <font color="Olive" title="b3">W</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> st <br/>( <font color="Olive" title="b3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">W</font> &amp; <font color="Olive" title="b3">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">A</font> &amp; <font color="Maroon" title="c4">Null</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b3">W</font> ) ) );<br/><span class="kw">assume </span><a NAME="E1:63_1"/><span class="lab"><font color="Green" title="E39">A2</font></span>: 
( <font color="Maroon" title="c4">Null</font> is  <a href="topdim_1.html#NV2" title="TOPDIM_1:NV.2">finite-ind</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Maroon" title="c4">Null</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">RNG</font> being   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c1">TM</font> st <br/> for <font color="Olive" title="b2">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font>  st <font color="Olive" title="b2">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">RNG</font> holds <br/><font color="Maroon" title="c4">Null</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive">b<sub>4</sub></font></span><br/><a NAME="E2:63_1"/><span class="lab"><font color="Green" title="E40">A3</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">TM</font></span>)</span>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">TM</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/> ex <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">TM</font></span>)</span>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">TM</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> implies  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:63_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">TM</font></span>)</span>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">TM</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">p9</font>, <font color="Maroon" title="c7">A9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:63_1_1"/><span class="lab"><font color="Green" title="E41">A4</font></span>: 
<font color="Maroon" title="c6">p9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">TM</font>
 <span class="kw">and </span><br/><a NAME="E4:63_1_1"/><span class="lab"><font color="Green" title="E42">A5</font></span>: 
<font color="Maroon" title="c7">A9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">TM</font>
 <span class="kw">and </span><br/><a NAME="E5:63_1_1"/><span class="lab"><font color="Green" title="E43">A6</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">p9</font>,<font color="Maroon" title="c7">A9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">p9</font> = <font color="Maroon" title="c6">p9</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">TM</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_1_1"><span class="lab"><font color="Green" title="E41">A4</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">A9</font> = <font color="Maroon" title="c7">A9</font> as   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63_1_1"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="ref" href="pre_topc.html#D2" onmouseover="rs('pre_topc/D2')" onmouseout="rh()">PRE_TOPC:def 2</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:63_1_1_1"/>
(  not <font color="Maroon" title="c8">p9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">A9</font> or <font color="Maroon" title="c8">p9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">A9</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:63_1_1_1_1"/><span class="lab"><font color="Green" title="E44">A7</font></span>: 
 not <font color="Maroon" title="c8">p9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">A9</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span>
 <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">TM</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>, <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">TM</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c10">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">TM</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font>  st <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">p</font>,<font color="Olive" title="b1">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>(  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">TM</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">TM</font> &amp; (  not <font color="Maroon" title="c10">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">A</font> implies  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">TM</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">TM</font> ) &amp; ( <font color="Maroon" title="c10">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">A</font> implies  ex <font color="Olive" title="b2">W</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> st <br/>( <font color="Olive" title="b2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">TM</font> &amp; <font color="Maroon" title="c10">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">W</font> &amp; <font color="Olive" title="b2">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> &amp; <font color="Maroon" title="c4">Null</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b2">W</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c11">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">p</font>,<font color="Maroon" title="c11">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> implies (  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">TM</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">TM</font> &amp; (  not <font color="Maroon" title="c10">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">A</font> implies  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">TM</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">TM</font> ) &amp; ( <font color="Maroon" title="c10">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">A</font> implies  ex <font color="Olive" title="b1">W</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> st <br/>( <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">TM</font> &amp; <font color="Maroon" title="c10">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">A</font> &amp; <font color="Maroon" title="c4">Null</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b1">W</font> ) ) ) )</span><br/><span class="kw">assume </span><a NAME="E2:63_1_1_1_1"/><span class="lab"><font color="Green" title="E45">A8</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">p</font>,<font color="Maroon" title="c11">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">TM</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">TM</font> &amp; (  not <font color="Maroon" title="c10">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">A</font> implies  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">TM</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">TM</font> ) &amp; ( <font color="Maroon" title="c10">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">A</font> implies  ex <font color="Olive" title="b1">W</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> st <br/>( <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">TM</font> &amp; <font color="Maroon" title="c10">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">A</font> &amp; <font color="Maroon" title="c4">Null</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b1">W</font> ) ) )</span><br/><a NAME="E3:63_1_1_1_1"/>
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:63_1_1"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E2:63_1_1_1_1"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:63_1_1_1_1"/>
(  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">TM</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">TM</font> &amp; (  not <font color="Maroon" title="c10">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">A</font> implies  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">TM</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">TM</font> ) &amp; ( <font color="Maroon" title="c10">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">A</font> implies  ex <font color="Olive" title="b1">W</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> st <br/>( <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">TM</font> &amp; <font color="Maroon" title="c10">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">A</font> &amp; <font color="Maroon" title="c4">Null</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b1">W</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:63_1_1"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E1:63_1_1_1_1"><span class="lab"><font color="Green" title="E44">A7</font></span></a>, <a class="txt" href="#E2:63_1_1_1_1"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="ref" href="pre_topc.html#D2" onmouseover="rs('pre_topc/D2')" onmouseout="rh()">PRE_TOPC:def 2</a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:63_1_1_1_2"/>
<font color="Maroon" title="c8">p9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">A9</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">W</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font><span class="kw"> such that </span><br/><a NAME="E3:63_1_1_1_2"/><span class="lab"><font color="Green" title="E44">A9</font></span>: 
( <font color="Maroon" title="c8">p9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">W</font> &amp; <font color="Maroon" title="c10">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">A9</font> &amp; <font color="Maroon" title="c4">Null</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c10">W</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E1:63_1"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="ref" href="topdim_1.html#T38" target="_self" onmouseover="rs('topdim_1/T38')" onmouseout="rh()">Th38</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c10">W</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c10">W</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c11">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">TM</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font>  st <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">p</font>,<font color="Olive" title="b1">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( <font color="Maroon" title="c10">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">TM</font> &amp; (  not <font color="Maroon" title="c11">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">A</font> implies <font color="Maroon" title="c10">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">TM</font> ) &amp; ( <font color="Maroon" title="c11">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">A</font> implies  ex <font color="Olive" title="b2">W</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> st <br/>( <font color="Olive" title="b2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">W</font> &amp; <font color="Maroon" title="c11">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">W</font> &amp; <font color="Olive" title="b2">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> &amp; <font color="Maroon" title="c4">Null</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b2">W</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c12">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> implies ( <font color="Maroon" title="c10">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">TM</font> &amp; (  not <font color="Maroon" title="c11">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">A</font> implies <font color="Maroon" title="c10">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">TM</font> ) &amp; ( <font color="Maroon" title="c11">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">A</font> implies  ex <font color="Olive" title="b1">W</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> st <br/>( <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">W</font> &amp; <font color="Maroon" title="c11">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">A</font> &amp; <font color="Maroon" title="c4">Null</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b1">W</font> ) ) ) )</span><br/><span class="kw">assume </span><a NAME="E4:63_1_1_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">TM</font> &amp; (  not <font color="Maroon" title="c11">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">A</font> implies <font color="Maroon" title="c10">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">TM</font> ) &amp; ( <font color="Maroon" title="c11">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">A</font> implies  ex <font color="Olive" title="b1">W</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> st <br/>( <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">W</font> &amp; <font color="Maroon" title="c11">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">A</font> &amp; <font color="Maroon" title="c4">Null</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b1">W</font> ) ) )</span><br/><a NAME="E5:63_1_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p9</font> &amp; <font color="Maroon" title="c12">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:63_1_1"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:63_1_1_1_2"/>
( <font color="Maroon" title="c10">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">TM</font> &amp; (  not <font color="Maroon" title="c11">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">A</font> implies <font color="Maroon" title="c10">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">TM</font> ) &amp; ( <font color="Maroon" title="c11">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">A</font> implies  ex <font color="Olive" title="b1">W</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> st <br/>( <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">W</font> &amp; <font color="Maroon" title="c11">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">A</font> &amp; <font color="Maroon" title="c4">Null</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b1">W</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_1_1_1_2"><span class="lab"><font color="Green" title="E44">A9</font></span></a>, <a class="ref" href="pre_topc.html#D2" onmouseover="rs('pre_topc/D2')" onmouseout="rh()">PRE_TOPC:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c5">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E4:63_1"/><span class="lab"><font color="Green" title="E41">A10</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">TM</font></span>)</span>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">TM</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E5:63_1"/><span class="lab"><font color="Green" title="E42">A11</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">TM</font></span>)</span>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">TM</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c5">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="classes1.html#S1" onmouseover="rs('classes1/S1')" onmouseout="rh()">CLASSES1:sch 1</a>(<a class="txt" href="#E2:63_1"><span class="lab"><font color="Green" title="E40">A3</font></span></a>);<br/></span><a NAME="E6:63_1"/><span class="lab"><font color="Green" title="E43">A12</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c5">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">TM</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c5">f</font> or <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">TM</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:63_1_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c5">f</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">TM</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:63_1_2"/><span class="lab"><font color="Green" title="E44">A13</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">and </span><br/><a NAME="E4:63_1_2"/><span class="lab"><font color="Green" title="E45">A14</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E5:63_1_2"/>
 ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">TM</font> &amp; <font color="Olive" title="b2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">TM</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63_1"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E3:63_1_2"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:63_1_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">TM</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63_1"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E5:63_1"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="txt" href="#E3:63_1_2"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="txt" href="#E4:63_1_2"><span class="lab"><font color="Green" title="E45">A14</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">RNG</font> =  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c5">f</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">TM</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font>  st <font color="Olive" title="b1">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/> for <font color="Olive" title="b2">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">TM</font>  st <font color="Olive" title="b2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">A</font> holds <br/> ex <font color="Olive" title="b3">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> st <br/>( <font color="Olive" title="b3">W</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">RNG</font> &amp; <font color="Olive" title="b2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">W</font> &amp; <font color="Olive" title="b3">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies  for <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">TM</font>  st <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">A</font> holds <br/> ex <font color="Olive" title="b2">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> st <br/>( <font color="Olive" title="b2">W</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">RNG</font> &amp; <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">W</font> &amp; <font color="Olive" title="b2">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font> ) )</span><br/><span class="kw">assume </span><a NAME="E1:63_1_3"/>
<font color="Maroon" title="c7">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">TM</font>  st <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">A</font> holds <br/> ex <font color="Olive" title="b2">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> st <br/>( <font color="Olive" title="b2">W</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">RNG</font> &amp; <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">W</font> &amp; <font color="Olive" title="b2">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font> )</span><br/><a NAME="E2:63_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A15</font></span>: 
<font color="Maroon" title="c7">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">TM</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D2" onmouseover="rs('pre_topc/D2')" onmouseout="rh()">PRE_TOPC:def 2</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c8">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">TM</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">A</font> implies  ex <font color="Olive" title="b1">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> st <br/>( <font color="Olive" title="b1">W</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">RNG</font> &amp; <font color="Maroon" title="c8">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font> ) )</span><br/><span class="kw">assume </span><a NAME="E3:63_1_3"/><span class="lab"><font color="Green" title="E45">A16</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">A</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> st <br/>( <font color="Olive" title="b1">W</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">RNG</font> &amp; <font color="Maroon" title="c8">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font> )</span><br/><a NAME="E4:63_1_3"/><span class="lab"><font color="Green" title="E46">A17</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c7">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">TM</font></span>)</span>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">TM</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_1_3"><span class="lab"><font color="Green" title="E44">A15</font></span></a>, <a class="txt" href="#E3:63_1_3"><span class="lab"><font color="Green" title="E45">A16</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">W</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font><span class="kw"> such that </span><br/><a NAME="E6:63_1_3"/><span class="lab"><font color="Green" title="E47">A18</font></span>: 
( <font color="Maroon" title="c9">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c7">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c8">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">W</font> &amp; <font color="Maroon" title="c9">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font> )
 <span class="kw">and </span><br/><a NAME="E7:63_1_3"/>
<font color="Maroon" title="c4">Null</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c9">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:63_1"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="txt" href="#E3:63_1_3"><span class="lab"><font color="Green" title="E45">A16</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">W</font> = <font color="Maroon" title="c9">W</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> ;<br/><span class="kw">take </span><font color="Maroon" title="c11">W</font> = <font color="Maroon" title="c10">W</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">W</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">RNG</font> &amp; <font color="Maroon" title="c8">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">W</font> &amp; <font color="Maroon" title="c11">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font> )</span><br/><span class="kw">thus </span><a NAME="E9:63_1_3"/>
( <font color="Maroon" title="c11">W</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">RNG</font> &amp; <font color="Maroon" title="c8">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">W</font> &amp; <font color="Maroon" title="c11">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63_1"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E4:63_1_3"><span class="lab"><font color="Green" title="E46">A17</font></span></a>, <a class="txt" href="#E6:63_1_3"><span class="lab"><font color="Green" title="E47">A18</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">RNG</font> = <font color="Maroon" title="c6">RNG</font> as   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c1">TM</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:63_1"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="ref" href="yellow_9.html#T32" onmouseover="rs('yellow_9/T32')" onmouseout="rh()">YELLOW_9:32</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c8">RNG</font> = <font color="Maroon" title="c7">RNG</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font>  st <font color="Olive" title="b1">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">RNG</font> holds <br/><font color="Maroon" title="c4">Null</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive">b<sub>3</sub></font></span><br/><span class="kw">let </span><font color="Maroon" title="c9">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">RNG</font> implies <font color="Maroon" title="c4">Null</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive">b<sub>2</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E10:63_1"/>
<font color="Maroon" title="c9">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">RNG</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">Null</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive">b<sub>2</sub></font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E12:63_1"/><span class="lab"><font color="Green" title="E44">A19</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">and </span><br/><a NAME="E13:63_1"/><span class="lab"><font color="Green" title="E45">A20</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E15:63_1"/><span class="lab"><font color="Green" title="E46">A21</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">TM</font>
 <span class="kw">and </span><br/><a NAME="E16:63_1"/><span class="lab"><font color="Green" title="E47">A22</font></span>: 
<font color="Maroon" title="c12">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">TM</font>
 <span class="kw">and </span><br/><a NAME="E17:63_1"/><span class="lab"><font color="Green" title="E48">A23</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63_1"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E12:63_1"><span class="lab"><font color="Green" title="E44">A19</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">A</font> = <font color="Maroon" title="c12">A</font> as    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" onmouseover="rs('tarski/T1')" onmouseout="rh()">TARSKI:1</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_4"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:63_1_4"/>
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A</font> or  not <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_4_1"><span class="kw">suppose </span></a><a NAME="E1:63_1_4_1"/>
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">Null</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><a NAME="E2:63_1_4_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">W</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> st <br/>( <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">A</font> &amp; <font color="Maroon" title="c4">Null</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b1">W</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63_1"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E5:63_1"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="txt" href="#E12:63_1"><span class="lab"><font color="Green" title="E44">A19</font></span></a>, <a class="txt" href="#E16:63_1"><span class="lab"><font color="Green" title="E47">A22</font></span></a>, <a class="txt" href="#E17:63_1"><span class="lab"><font color="Green" title="E48">A23</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:63_1_4_1"/>
<font color="Maroon" title="c4">Null</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c9">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:63_1"><span class="lab"><font color="Green" title="E45">A20</font></span></a>, <a class="txt" href="#E17:63_1"><span class="lab"><font color="Green" title="E48">A23</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_4_2"><span class="kw">suppose </span></a><a NAME="E1:63_1_4_2"/>
 not <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">Null</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><a NAME="E2:63_1_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">TM</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63_1"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E5:63_1"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="txt" href="#E12:63_1"><span class="lab"><font color="Green" title="E44">A19</font></span></a>, <a class="txt" href="#E13:63_1"><span class="lab"><font color="Green" title="E45">A20</font></span></a>, <a class="txt" href="#E15:63_1"><span class="lab"><font color="Green" title="E46">A21</font></span></a>, <a class="txt" href="#E16:63_1"><span class="lab"><font color="Green" title="E47">A22</font></span></a>, <a class="txt" href="#E17:63_1"><span class="lab"><font color="Green" title="E48">A23</font></span></a></span>;<br/><a NAME="E3:63_1_4_2"/><span class="kw">then </span>
 <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c9">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">TM</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topgen_1.html#T14" onmouseover="rs('topgen_1/T14')" onmouseout="rh()">TOPGEN_1:14</a></span>;<br/><span class="kw">hence </span><a NAME="E4:63_1_4_2"/>
<font color="Maroon" title="c4">Null</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c9">B</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div>

<span class="kw">given </span><font color="Maroon" title="c5">B</font> being   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c1">TM</font><span class="kw"> such that </span><a NAME="E4:63"/><span class="lab"><font color="Green" title="E39">A24</font></span>: 
 for <font color="Olive" title="b1">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font>  st <font color="Olive" title="b1">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">B</font> holds <br/><font color="Maroon" title="c4">Null</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b1">A</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">Null</font> is  <a href="topdim_1.html#NV2" title="TOPDIM_1:NV.2">finite-ind</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Maroon" title="c4">Null</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )</span><br/>

<a NAME="E5:63"/>
 for <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">TM</font><br/>  for <font color="Olive" title="b2">U</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font>  st <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">U</font> holds <br/> ex <font color="Olive" title="b3">W</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> st <br/>( <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">W</font> &amp; <font color="Olive" title="b3">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">U</font> &amp; <font color="Maroon" title="c4">Null</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b3">W</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">TM</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">U</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font>  st <font color="Maroon" title="c6">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">U</font> holds <br/> ex <font color="Olive" title="b2">W</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> st <br/>( <font color="Maroon" title="c6">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">W</font> &amp; <font color="Olive" title="b2">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">U</font> &amp; <font color="Maroon" title="c4">Null</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b2">W</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c7">U</font> be   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">U</font> implies  ex <font color="Olive" title="b1">W</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> st <br/>( <font color="Maroon" title="c6">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">U</font> &amp; <font color="Maroon" title="c4">Null</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b1">W</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:63_2"/>
<font color="Maroon" title="c6">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">U</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> st <br/>( <font color="Maroon" title="c6">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">U</font> &amp; <font color="Maroon" title="c4">Null</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b1">W</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">a</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font><span class="kw"> such that </span><br/><a NAME="E3:63_2"/><span class="lab"><font color="Green" title="E40">A25</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">and </span><br/><a NAME="E4:63_2"/><span class="lab"><font color="Green" title="E41">A26</font></span>: 
( <font color="Maroon" title="c6">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c8">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">U</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#T31" onmouseover="rs('yellow_9/T31')" onmouseout="rh()">YELLOW_9:31</a></span>;<br/>
<a NAME="E5:63_2"/>
<font color="Maroon" title="c5">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">TM</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_2.html#T64" onmouseover="rs('tops_2/T64')" onmouseout="rh()">TOPS_2:64</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c8">a</font> as   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_2"><span class="lab"><font color="Green" title="E40">A25</font></span></a>, <a class="ref" href="pre_topc.html#D2" onmouseover="rs('pre_topc/D2')" onmouseout="rh()">PRE_TOPC:def 2</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c9">a</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">a</font> &amp; <font color="Maroon" title="c9">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">U</font> &amp; <font color="Maroon" title="c4">Null</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c9">a</font> )</span><br/>

<span class="kw">thus </span><a NAME="E7:63_2"/>
( <font color="Maroon" title="c6">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">a</font> &amp; <font color="Maroon" title="c9">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">U</font> &amp; <font color="Maroon" title="c4">Null</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c9">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63"><span class="lab"><font color="Green" title="E39">A24</font></span></a>, <a class="txt" href="#E3:63_2"><span class="lab"><font color="Green" title="E40">A25</font></span></a>, <a class="txt" href="#E4:63_2"><span class="lab"><font color="Green" title="E41">A26</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:63"/>
( <font color="Maroon" title="c4">Null</font> is  <a href="topdim_1.html#NV2" title="TOPDIM_1:NV.2">finite-ind</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Maroon" title="c4">Null</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="topdim_1.html#T38" target="_self" onmouseover="rs('topdim_1/T38')" onmouseout="rh()">Th38</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
