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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">iC</font> being   <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>  <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a><br/>  for <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 (  for <font color="Olive" title="b3">Fm</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">TM</font>  st <font color="Olive" title="b3">Fm</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  &amp; <font color="Olive" title="b3">Fm</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c1">TM</font> holds <br/> ex <font color="Olive" title="b4">Gm</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">TM</font> st <br/>( <font color="Olive" title="b4">Gm</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">Fm</font> &amp; <font color="Olive" title="b4">Gm</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c1">TM</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b4">Gm</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">iC</font> ) ) holds <br/> ex <font color="Olive" title="b3">underB</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/>( <font color="Olive" title="b3">underB</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">B</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b3">underB</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">iC</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">iC</font> be   <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>  <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</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">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 (  for <font color="Olive" title="b2">Fm</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">TM</font>  st <font color="Olive" title="b2">Fm</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  &amp; <font color="Olive" title="b2">Fm</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c1">TM</font> holds <br/> ex <font color="Olive" title="b3">Gm</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">TM</font> st <br/>( <font color="Olive" title="b3">Gm</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Fm</font> &amp; <font color="Olive" title="b3">Gm</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c1">TM</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b3">Gm</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">iC</font> ) ) holds <br/> ex <font color="Olive" title="b2">underB</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/>( <font color="Olive" title="b2">underB</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b2">underB</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">iC</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">B</font> be   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</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">Fm</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">TM</font>  st <font color="Olive" title="b1">Fm</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  &amp; <font color="Olive" title="b1">Fm</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c1">TM</font> holds <br/> ex <font color="Olive" title="b2">Gm</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">TM</font> st <br/>( <font color="Olive" title="b2">Gm</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Fm</font> &amp; <font color="Olive" title="b2">Gm</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c1">TM</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b2">Gm</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">iC</font> ) ) implies  ex <font color="Olive" title="b1">underB</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/>( <font color="Olive" title="b1">underB</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">underB</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">iC</font> ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:39"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
 for <font color="Olive" title="b1">F</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">TM</font>  st <font color="Olive" title="b1">F</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  &amp; <font color="Olive" title="b1">F</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c1">TM</font> holds <br/> ex <font color="Olive" title="b2">G</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">TM</font> st <br/>( <font color="Olive" title="b2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">F</font> &amp; <font color="Olive" title="b2">G</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c1">TM</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b2">G</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">iC</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">underB</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/>( <font color="Olive" title="b1">underB</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">underB</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">iC</font> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_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="E1:39_1"/>
( <font color="Maroon" title="c1">TM</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  or  not <font color="Maroon" title="c1">TM</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1"><span class="kw">suppose </span></a><a NAME="E1:39_1_1"/>
<font color="Maroon" title="c1">TM</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</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">underB</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/>( <font color="Olive" title="b1">underB</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">underB</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">iC</font> )</span><br/><div class="add"><a NAME="E2:39_1_1"/><span class="kw">then </span>
 <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <font color="Maroon" title="c1">TM</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">underB</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><br/><a NAME="E4:39_1_1"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c4">underB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel23.html#T74" onmouseover="rs('waybel23/T74')" onmouseout="rh()">WAYBEL23:74</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c4">underB</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">underB</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c4">underB</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">iC</font> )</span><br/><a NAME="E5:39_1_1"/>
<font color="Maroon" title="c4">underB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:39_1_1"/>
( <font color="Maroon" title="c4">underB</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c4">underB</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">iC</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2"><span class="kw">suppose </span></a><a NAME="E1:39_1_2"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
 not <font color="Maroon" title="c1">TM</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</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">underB</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/>( <font color="Olive" title="b1">underB</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">underB</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">iC</font> )</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c4">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">set </span><font color="Maroon" title="c5">cT</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">TM</font>;<br/><span class="kw">consider </span><font color="Maroon" title="c6">metr</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">TM</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">TM</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E3:39_1_2"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
<font color="Maroon" title="c6">metr</font> <a href="pcomps_1.html#R1" title="PCOMPS_1:pred.1">is_metric_of</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">TM</font>
 <span class="kw">and </span><br/><a NAME="E4:39_1_2"/><span class="lab"><font color="Green" title="E29">A5</font></span>: 
 <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">Family_open_set</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">SpaceMetr</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">TM</font>,<font color="Maroon" title="c6">metr</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="pcomps_1.html#D8" onmouseover="rs('pcomps_1/D8')" onmouseout="rh()">PCOMPS_1:def 8</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">Tm</font> =  <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">SpaceMetr</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">TM</font>,<font color="Maroon" title="c6">metr</font>) as   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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1_2"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E3:39_1_2"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="ref" href="pcomps_1.html#T36" onmouseover="rs('pcomps_1/T36')" onmouseout="rh()">PCOMPS_1:36</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">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> holds <br/> ex <font color="Olive" title="b2">G</font> being   <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">TM</font> st <br/>( <font color="Olive" title="b2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b3">U</font> where <font color="Olive" title="b3">U</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> : ( <font color="Olive" title="b3">U</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp;  ex <font color="Olive" title="b4">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c7">Tm</font> st <font color="Olive" title="b3">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b4">p</font>,<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span>) ) </span> } </span>  &amp; <font color="Olive" title="b2">G</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c1">TM</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b2">G</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">iC</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">G</font> );<br/><a NAME="E6:39_1_2"/><span class="lab"><font color="Green" title="E30">A6</font></span>: 
<font color="Maroon" title="c3">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/><a NAME="E7:39_1_2"/><span class="lab"><font color="Green" title="E31">A7</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>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  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="39_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">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="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  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="c8">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:39_1_2_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</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">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="c8">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="c9">n</font> = <font color="Maroon" title="c8">x</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">F</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">U</font> where <font color="Olive" title="b1">U</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> : ( <font color="Olive" title="b1">U</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp;  ex <font color="Olive" title="b2">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">Tm</font> st <font color="Olive" title="b1">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b2">p</font>,<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span>) ) </span> } </span> ;<br/>
<a NAME="E3:39_1_2_1"/><span class="lab"><font color="Green" title="E32">A8</font></span>: 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">U</font> where <font color="Olive" title="b1">U</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> : ( <font color="Olive" title="b1">U</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp;  ex <font color="Olive" title="b2">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">Tm</font> st <font color="Olive" title="b1">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b2">p</font>,<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span>) ) </span> } </span>  <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="39_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">f</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="c11">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">U</font> where <font color="Olive" title="b1">U</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> : ( <font color="Olive" title="b1">U</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp;  ex <font color="Olive" title="b2">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">Tm</font> st <font color="Olive" title="b1">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b2">p</font>,<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span>) ) </span> } </span>  or <font color="Maroon" title="c11">f</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="E1:39_1_2_1_1"/>
<font color="Maroon" title="c11">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">U</font> where <font color="Olive" title="b1">U</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> : ( <font color="Olive" title="b1">U</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp;  ex <font color="Olive" title="b2">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">Tm</font> st <font color="Olive" title="b1">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b2">p</font>,<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span>) ) </span> } </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="c11">f</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/>

<a NAME="E2:39_1_2_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">U</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="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">U</font> &amp; <font color="Olive" title="b1">U</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp;  ex <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="c7">Tm</font> st <font color="Olive" title="b1">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b2">p</font>,<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span>) )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:39_1_2_1_1"/>
<font color="Maroon" title="c11">f</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="#E6:39_1_2"><span class="lab"><font color="Green" title="E30">A6</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="c11">F</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">U</font> where <font color="Olive" title="b1">U</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> : ( <font color="Olive" title="b1">U</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp;  ex <font color="Olive" title="b2">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">Tm</font> st <font color="Olive" title="b1">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b2">p</font>,<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span>) ) </span> } </span>  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/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">F</font> = <font color="Maroon" title="c11">F</font> as   <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  <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="txt" href="#E3:39_1_2_1"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="ref" href="tops_2.html#T11" onmouseover="rs('tops_2/T11')" onmouseout="rh()">TOPS_2:11</a></span>;<br/>
<a NAME="E6:39_1_2_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">TM</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c12">F</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="39_1_2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">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="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">TM</font> or <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c12">F</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:39_1_2_1_2"/><span class="lab"><font color="Green" title="E33">A9</font></span>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c12">F</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">p</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">TM</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">q</font> = <font color="Maroon" title="c13">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">Tm</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1_2"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E3:39_1_2"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E1:39_1_2_1_2"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="ref" href="pcomps_2.html#T4" onmouseover="rs('pcomps_2/T4')" onmouseout="rh()">PCOMPS_2:4</a></span>;<br/>
<a NAME="E4:39_1_2_1_2"/>
( 2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c15">q</font>,<font color="Maroon" title="c15">q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T1" onmouseover="rs('metric_1/T1')" onmouseout="rh()">METRIC_1:1</a>, <a class="ref" href="newton.html#T83" onmouseover="rs('newton/T83')" onmouseout="rh()">NEWTON:83</a></span>;<br/>
<a NAME="E5:39_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A10</font></span>: 
<font color="Maroon" title="c15">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c15">q</font>,<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T11" onmouseover="rs('metric_1/T11')" onmouseout="rh()">METRIC_1:11</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">BALL</font> =  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c15">q</font>,<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span>) as   <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="#E1:39_1_2"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E3:39_1_2"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="ref" href="pcomps_2.html#T4" onmouseover="rs('pcomps_2/T4')" onmouseout="rh()">PCOMPS_2:4</a></span>;<br/>
<a NAME="E7:39_1_2_1_2"/>
<font color="Maroon" title="c16">BALL</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">Family_open_set</a> <font color="Maroon" title="c7">Tm</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pcomps_1.html#T29" onmouseover="rs('pcomps_1/T29')" onmouseout="rh()">PCOMPS_1:29</a></span>;<br/>
<a NAME="E8:39_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">BALL</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1_2"><span class="lab"><font color="Green" title="E29">A5</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">U</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="E10:39_1_2_1_2"/><span class="lab"><font color="Green" title="E35">A11</font></span>: 
<font color="Maroon" title="c17">U</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 <span class="kw">and </span><br/><a NAME="E11:39_1_2_1_2"/><span class="lab"><font color="Green" title="E36">A12</font></span>: 
<font color="Maroon" title="c14">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c17">U</font>
 <span class="kw">and </span><br/><a NAME="E12:39_1_2_1_2"/><span class="lab"><font color="Green" title="E37">A13</font></span>: 
<font color="Maroon" title="c17">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">BALL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39_1_2_1_2"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="ref" href="yellow_9.html#T31" onmouseover="rs('yellow_9/T31')" onmouseout="rh()">YELLOW_9:31</a></span>;<br/>
<a NAME="E13:39_1_2_1_2"/>
<font color="Maroon" title="c17">U</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:39_1_2_1_2"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="txt" href="#E12:39_1_2_1_2"><span class="lab"><font color="Green" title="E37">A13</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:39_1_2_1_2"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c12">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:39_1_2_1_2"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:39_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">F</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c1">TM</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#D11" onmouseover="rs('setfam_1/D11')" onmouseout="rh()">SETFAM_1:def 11</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">G</font> being   <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"> such that </span><br/><a NAME="E9:39_1_2_1"/><span class="lab"><font color="Green" title="E33">A14</font></span>: 
<font color="Maroon" title="c13">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">F</font>
 <span class="kw">and </span><br/><a NAME="E10:39_1_2_1"/><span class="lab"><font color="Green" title="E34">A15</font></span>: 
( <font color="Maroon" title="c13">G</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c1">TM</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c13">G</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">iC</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E26">A1</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c13">G</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="c8">x</font>,<font color="Maroon" title="c13">G</font>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c14">m</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">m</font> implies  ex <font color="Olive" title="b1">G</font> being   <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">TM</font> st <br/>( <font color="Olive" title="b1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">U</font> where <font color="Olive" title="b2">U</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> : ( <font color="Olive" title="b2">U</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp;  ex <font color="Olive" title="b3">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c7">Tm</font> st <font color="Olive" title="b2">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b3">p</font>,<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c14">m</font></span>)</span></span>)</span>) ) </span> } </span>  &amp; <font color="Olive" title="b1">G</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c1">TM</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">G</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">iC</font> &amp; <font color="Maroon" title="c13">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">G</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E11:39_1_2_1"/><span class="lab"><font color="Green" title="E35">A16</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">m</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">G</font> being   <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">TM</font> st <br/>( <font color="Olive" title="b1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">U</font> where <font color="Olive" title="b2">U</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> : ( <font color="Olive" title="b2">U</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp;  ex <font color="Olive" title="b3">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c7">Tm</font> st <font color="Olive" title="b2">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b3">p</font>,<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c14">m</font></span>)</span></span>)</span>) ) </span> } </span>  &amp; <font color="Olive" title="b1">G</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c1">TM</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">G</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">iC</font> &amp; <font color="Maroon" title="c13">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">G</font> )</span><br/>

<a NAME="E12:39_1_2_1"/>
<font color="Maroon" title="c13">G</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39_1_2_1"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="ref" href="tops_2.html#T11" onmouseover="rs('tops_2/T11')" onmouseout="rh()">TOPS_2:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:39_1_2_1"/>
 ex <font color="Olive" title="b1">G</font> being   <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">TM</font> st <br/>( <font color="Olive" title="b1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">U</font> where <font color="Olive" title="b2">U</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> : ( <font color="Olive" title="b2">U</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp;  ex <font color="Olive" title="b3">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c7">Tm</font> st <font color="Olive" title="b2">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b3">p</font>,<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c14">m</font></span>)</span></span>)</span>) ) </span> } </span>  &amp; <font color="Olive" title="b1">G</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c1">TM</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">G</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">iC</font> &amp; <font color="Maroon" title="c13">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39_1_2_1"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="txt" href="#E10:39_1_2_1"><span class="lab"><font color="Green" title="E34">A15</font></span></a>, <a class="txt" href="#E11:39_1_2_1"><span class="lab"><font color="Green" title="E35">A16</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">consider </span><font color="Maroon" title="c8">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="E9:39_1_2"/><span class="lab"><font color="Green" title="E32">A17</font></span>: 
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp; (  for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</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="#E7:39_1_2"><span class="lab"><font color="Green" title="E31">A7</font></span></a>);<br/></span><a NAME="E10:39_1_2"/><span class="lab"><font color="Green" title="E33">A18</font></span>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="39_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">b</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="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c8">f</font></span>)</span> or <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:39_1_2_2"/>
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c8">f</font></span>)</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="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:39_1_2_2"/><span class="lab"><font color="Green" title="E34">A19</font></span>: 
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">y</font>
 <span class="kw">and </span><br/><a NAME="E4:39_1_2_2"/><span class="lab"><font color="Green" title="E35">A20</font></span>: 
<font color="Maroon" title="c10">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c8">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:39_1_2_2"/><span class="lab"><font color="Green" title="E36">A21</font></span>: 
<font color="Maroon" title="c11">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="c8">f</font>
 <span class="kw">and </span><br/><a NAME="E7:39_1_2_2"/><span class="lab"><font color="Green" title="E37">A22</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1_2_2"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <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">reconsider </span><font color="Maroon" title="c12">n</font> = <font color="Maroon" title="c11">x</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39_1_2"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E6:39_1_2_2"><span class="lab"><font color="Green" title="E36">A21</font></span></a></span>;<br/>
<a NAME="E9:39_1_2_2"/>
 ex <font color="Olive" title="b1">G</font> being   <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">TM</font> st <br/>( <font color="Olive" title="b1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">U</font> where <font color="Olive" title="b2">U</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> : ( <font color="Olive" title="b2">U</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp;  ex <font color="Olive" title="b3">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c7">Tm</font> st <font color="Olive" title="b2">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b3">p</font>,<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c12">n</font></span>)</span></span>)</span>) ) </span> } </span>  &amp; <font color="Olive" title="b1">G</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c1">TM</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">G</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">iC</font> &amp; <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">G</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39_1_2"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E6:39_1_2_2"><span class="lab"><font color="Green" title="E36">A21</font></span></a></span>;<br/>
<a NAME="E10:39_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">U</font> where <font color="Olive" title="b1">U</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> : ( <font color="Olive" title="b1">U</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp;  ex <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="c7">Tm</font> st <font color="Olive" title="b1">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b2">p</font>,<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c12">n</font></span>)</span></span>)</span>) ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_1_2_2"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="txt" href="#E7:39_1_2_2"><span class="lab"><font color="Green" title="E37">A22</font></span></a></span>;<br/>
<a NAME="E11:39_1_2_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">U</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">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> &amp; <font color="Olive" title="b1">U</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp;  ex <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="c7">Tm</font> st <font color="Olive" title="b1">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b2">p</font>,<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c12">n</font></span>)</span></span>)</span>) )
 
;<br/>
<span class="kw">hence </span><a NAME="E12:39_1_2_2"/>
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">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><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">Urngf</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c8">f</font></span>)</span> 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/><a NAME="E12:39_1_2"/>
 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">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 <br/>( <font color="Olive" title="b3">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">Urngf</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">a</font> &amp; <font color="Olive" title="b3">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="39_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">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="c10">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="c10">A</font> holds <br/> ex <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 <br/>( <font color="Olive" title="b2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">Urngf</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> &amp; <font color="Olive" title="b2">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">A</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:39_1_2_3"/>
<font color="Maroon" title="c10">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="c10">A</font> holds <br/> ex <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 <br/>( <font color="Olive" title="b2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">Urngf</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> &amp; <font color="Olive" title="b2">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">A</font> )</span><br/>

<a NAME="E2:39_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A23</font></span>: 
<font color="Maroon" title="c10">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">Family_open_set</a> <font color="Maroon" title="c7">Tm</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1_2"><span class="lab"><font color="Green" title="E29">A5</font></span></a></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"> ( <font color="Maroon" title="c11">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">A</font> implies  ex <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 <br/>( <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">Urngf</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> &amp; <font color="Olive" title="b1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">A</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E3:39_1_2_3"/><span class="lab"><font color="Green" title="E35">A24</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">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">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 <br/>( <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">Urngf</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> &amp; <font color="Olive" title="b1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">A</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c12">p9</font> = <font color="Maroon" title="c11">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">Tm</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1_2"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E3:39_1_2"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="ref" href="pcomps_2.html#T4" onmouseover="rs('pcomps_2/T4')" onmouseout="rh()">PCOMPS_2:4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E6:39_1_2_3"/><span class="lab"><font color="Green" title="E36">A25</font></span>: 
<font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E7:39_1_2_3"/><span class="lab"><font color="Green" title="E37">A26</font></span>: 
 <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c12">p9</font>,<font color="Maroon" title="c13">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_1_2_3"><span class="lab"><font color="Green" title="E34">A23</font></span></a>, <a class="txt" href="#E3:39_1_2_3"><span class="lab"><font color="Green" title="E35">A24</font></span></a>, <a class="ref" href="pcomps_1.html#D4" onmouseover="rs('pcomps_1/D4')" onmouseout="rh()">PCOMPS_1:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E9:39_1_2_3"/><span class="lab"><font color="Green" title="E38">A27</font></span>: 
1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c14">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_1_2_3"><span class="lab"><font color="Green" title="E36">A25</font></span></a>, <a class="ref" href="prepower.html#T92" onmouseover="rs('prepower/T92')" onmouseout="rh()">PREPOWER:92</a></span>;<br/>
<a NAME="E10:39_1_2_3"/><span class="lab"><font color="Green" title="E39">A28</font></span>: 
<font color="Maroon" title="c14">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">G</font> being   <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  <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"> such that </span><br/><a NAME="E12:39_1_2_3"/><span class="lab"><font color="Green" title="E40">A29</font></span>: 
<font color="Maroon" title="c15">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">U</font> where <font color="Olive" title="b1">U</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> : ( <font color="Olive" title="b1">U</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp;  ex <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="c7">Tm</font> st <font color="Olive" title="b1">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b2">p</font>,<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c14">n</font></span>)</span></span>)</span>) ) </span> } </span> 
 <span class="kw">and </span><br/><a NAME="E13:39_1_2_3"/><span class="lab"><font color="Green" title="E41">A30</font></span>: 
<font color="Maroon" title="c15">G</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c1">TM</font>
 <span class="kw">and </span><br/><a NAME="E14:39_1_2_3"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c15">G</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">iC</font>
 <span class="kw">and </span><br/><a NAME="E15:39_1_2_3"/><span class="lab"><font color="Green" title="E42">A31</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39_1_2"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E10:39_1_2_3"><span class="lab"><font color="Green" title="E39">A28</font></span></a></span>;<br/>
<a NAME="E16:39_1_2_3"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">TM</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c15">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:39_1_2_3"><span class="lab"><font color="Green" title="E41">A30</font></span></a>, <a class="ref" href="setfam_1.html#T45" onmouseover="rs('setfam_1/T45')" onmouseout="rh()">SETFAM_1:45</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">a</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E18:39_1_2_3"/><span class="lab"><font color="Green" title="E43">A32</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c16">a</font>
 <span class="kw">and </span><br/><a NAME="E19:39_1_2_3"/><span class="lab"><font color="Green" title="E44">A33</font></span>: 
<font color="Maroon" title="c16">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c15">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1_2"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<a NAME="E20:39_1_2_3"/>
<font color="Maroon" title="c16">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">U</font> where <font color="Olive" title="b1">U</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> : ( <font color="Olive" title="b1">U</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp;  ex <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="c7">Tm</font> st <font color="Olive" title="b1">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b2">p</font>,<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c14">n</font></span>)</span></span>)</span>) ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:39_1_2_3"><span class="lab"><font color="Green" title="E40">A29</font></span></a>, <a class="txt" href="#E19:39_1_2_3"><span class="lab"><font color="Green" title="E44">A33</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">U</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="E22:39_1_2_3"/><span class="lab"><font color="Green" title="E45">A34</font></span>: 
<font color="Maroon" title="c16">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">U</font>
 <span class="kw">and </span><br/><a NAME="E23:39_1_2_3"/>
<font color="Maroon" title="c17">U</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 <span class="kw">and </span><br/><a NAME="E24:39_1_2_3"/><span class="lab"><font color="Green" title="E46">A35</font></span>: 
 ex <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="c7">Tm</font> st <font color="Maroon" title="c17">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b1">p</font>,<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c14">n</font></span>)</span></span>)</span>)
 ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c17">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"> ( <font color="Maroon" title="c17">U</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">Urngf</font> &amp; <font color="Maroon" title="c11">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c17">U</font> &amp; <font color="Maroon" title="c17">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">A</font> )</span><br/>

<a NAME="E25:39_1_2_3"/>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c8">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39_1_2"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a>, <a class="txt" href="#E10:39_1_2_3"><span class="lab"><font color="Green" title="E39">A28</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E26:39_1_2_3"/>
( <font color="Maroon" title="c17">U</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">Urngf</font> &amp; <font color="Maroon" title="c11">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c17">U</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:39_1_2_3"><span class="lab"><font color="Green" title="E42">A31</font></span></a>, <a class="txt" href="#E18:39_1_2_3"><span class="lab"><font color="Green" title="E43">A32</font></span></a>, <a class="txt" href="#E19:39_1_2_3"><span class="lab"><font color="Green" title="E44">A33</font></span></a>, <a class="txt" href="#E22:39_1_2_3"><span class="lab"><font color="Green" title="E45">A34</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 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="c17">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">A</font></span><br/>

<span class="kw">thus </span><a NAME="E27:39_1_2_3"/>
<font color="Maroon" title="c17">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">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"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="39_1_2_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">u9</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="c18">u9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">U</font> or <font color="Maroon" title="c18">u9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c19">pp</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">Tm</font><span class="kw"> such that </span><br/><a NAME="E2:39_1_2_3_1"/><span class="lab"><font color="Green" title="E47">A36</font></span>: 
<font color="Maroon" title="c17">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c19">pp</font>,<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c14">n</font></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:39_1_2_3"><span class="lab"><font color="Green" title="E46">A35</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:39_1_2_3_1"/><span class="lab"><font color="Green" title="E48">A37</font></span>: 
<font color="Maroon" title="c18">u9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">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"> <font color="Maroon" title="c18">u9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">u</font> = <font color="Maroon" title="c18">u9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">Tm</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1_2"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E3:39_1_2"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="ref" href="pcomps_2.html#T4" onmouseover="rs('pcomps_2/T4')" onmouseout="rh()">PCOMPS_2:4</a></span>;<br/>
<a NAME="E5:39_1_2_3_1"/>
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c19">pp</font>,<font color="Maroon" title="c20">u</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c14">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_1_2_3_1"><span class="lab"><font color="Green" title="E47">A36</font></span></a>, <a class="txt" href="#E3:39_1_2_3_1"><span class="lab"><font color="Green" title="E48">A37</font></span></a>, <a class="ref" href="metric_1.html#T11" onmouseover="rs('metric_1/T11')" onmouseout="rh()">METRIC_1:11</a></span>;<br/>
<a NAME="E6:39_1_2_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A38</font></span>: 
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c19">pp</font>,<font color="Maroon" title="c20">u</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39_1_2_3"><span class="lab"><font color="Green" title="E38">A27</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E7:39_1_2_3_1"/>
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c19">pp</font>,<font color="Maroon" title="c12">p9</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c14">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:39_1_2_3"><span class="lab"><font color="Green" title="E43">A32</font></span></a>, <a class="txt" href="#E22:39_1_2_3"><span class="lab"><font color="Green" title="E45">A34</font></span></a>, <a class="txt" href="#E2:39_1_2_3_1"><span class="lab"><font color="Green" title="E47">A36</font></span></a>, <a class="ref" href="metric_1.html#T11" onmouseover="rs('metric_1/T11')" onmouseout="rh()">METRIC_1:11</a></span>;<br/>
<a NAME="E8:39_1_2_3_1"/><span class="kw">then </span>
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c12">p9</font>,<font color="Maroon" title="c19">pp</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39_1_2_3"><span class="lab"><font color="Green" title="E38">A27</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E9:39_1_2_3_1"/><span class="kw">then </span>
(  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c12">p9</font>,<font color="Maroon" title="c20">u</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c12">p9</font>,<font color="Maroon" title="c19">pp</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c19">pp</font>,<font color="Maroon" title="c20">u</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c12">p9</font>,<font color="Maroon" title="c19">pp</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c19">pp</font>,<font color="Maroon" title="c20">u</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_1_2_3_1"><span class="lab"><font color="Green" title="E49">A38</font></span></a>, <a class="ref" href="metric_1.html#T4" onmouseover="rs('metric_1/T4')" onmouseout="rh()">METRIC_1:4</a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/>
<a NAME="E10:39_1_2_3_1"/><span class="kw">then </span>
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c12">p9</font>,<font color="Maroon" title="c20">u</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E11:39_1_2_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c12">p9</font>,<font color="Maroon" title="c13">r</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T11" onmouseover="rs('metric_1/T11')" onmouseout="rh()">METRIC_1:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:39_1_2_3_1"/>
<font color="Maroon" title="c18">u9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:39_1_2_3"><span class="lab"><font color="Green" title="E37">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>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">Urngf</font> = <font color="Maroon" title="c9">Urngf</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="#E6:39_1_2"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E10:39_1_2"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</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="c10">Urngf</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="c10">Urngf</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c10">Urngf</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">iC</font> )</span><br/><a NAME="E14:39_1_2"/>
 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>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c8">f</font> holds <br/> <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">iC</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="39_1_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">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="c11">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="c8">f</font> implies  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">iC</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:39_1_2_4"/>
<font color="Maroon" title="c11">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="c8">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">  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">iC</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">n</font> = <font color="Maroon" title="c11">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39_1_2"><span class="lab"><font color="Green" title="E32">A17</font></span></a></span>;<br/>
<a NAME="E3:39_1_2_4"/>
 ex <font color="Olive" title="b1">G</font> being   <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">TM</font> st <br/>( <font color="Olive" title="b1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">U</font> where <font color="Olive" title="b2">U</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> : ( <font color="Olive" title="b2">U</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp;  ex <font color="Olive" title="b3">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c7">Tm</font> st <font color="Olive" title="b2">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b3">p</font>,<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <font color="Maroon" title="c12">n</font></span>)</span></span>)</span>) ) </span> } </span>  &amp; <font color="Olive" title="b1">G</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c1">TM</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">G</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">iC</font> &amp; <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">G</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39_1_2"><span class="lab"><font color="Green" title="E32">A17</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:39_1_2_4"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">iC</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><a NAME="E15:39_1_2"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c2">iC</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39_1_2"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="ref" href="card_1.html#T47" onmouseover="rs('card_1/T47')" onmouseout="rh()">CARD_1:47</a>, <a class="ref" href="card_2.html#T86" onmouseover="rs('card_2/T86')" onmouseout="rh()">CARD_2:86</a></span>;<br/><span class="kw">hence </span><a NAME="E16:39_1_2"/>
( <font color="Maroon" title="c10">Urngf</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c10">Urngf</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">iC</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:39_1_2"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="txt" href="metrizts.html#E18"><span class="lab"><font color="Green" title="E12">Lm5</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><span class="kw">end;</span></div>

</div>
