<?xml version="1.0"?>
<div class="add">

<a NAME="E1:22"/>
 for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a>  st  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> &amp; <font color="Olive" title="b2">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> &amp; <font color="Olive" title="b1">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  &amp; <font color="Olive" title="b2">B</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  holds <br/><font color="Olive" title="b1">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b2">B</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</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"> (  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> &amp; <font color="Maroon" title="c1">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  &amp; <font color="Maroon" title="c2">B</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  implies <font color="Maroon" title="c1">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">B</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:22_1"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">B</font>
 <span class="kw">and </span><br/><a NAME="E2:22_1"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<font color="Maroon" title="c1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a>
 <span class="kw">and </span><br/><a NAME="E3:22_1"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
<font color="Maroon" title="c2">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a>
 <span class="kw">and </span><br/><a NAME="E4:22_1"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
<font color="Maroon" title="c1">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">and </span><br/><a NAME="E5:22_1"/><span class="lab"><font color="Green" title="E21">A5</font></span>: 
<font color="Maroon" title="c2">B</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</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="c1">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">B</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">P</font> = <font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c4">Q</font> = <font color="Maroon" title="c2">B</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E7:22_1"/><span class="lab"><font color="Green" title="E22">A6</font></span>: 
<font color="Maroon" title="c1">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">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"> contradiction</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">x</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">P</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">x</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">P</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<div><span class="lab"><font color="Green" title="E23">A7</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font></span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c6">x</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font></span><br/><span class="kw">thus </span><a NAME="E1:22_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_1"><span class="lab"><font color="Green" title="E18">A2</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">set </span><font color="Maroon" title="c7">x</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">Q</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">x</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">Q</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<div><span class="lab"><font color="Green" title="E24">A8</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font></span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c8">x</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font></span><br/><span class="kw">thus </span><a NAME="E1:22_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_1"><span class="lab"><font color="Green" title="E19">A3</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>
<a NAME="E12:22_1"/><span class="lab"><font color="Green" title="E25">A9</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="metric_1.html#K7" title="METRIC_1:func.7">RealSpace</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <a href="metric_1.html#K7" title="METRIC_1:func.7">RealSpace</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T12" onmouseover="rs('topmetr/T12')" onmouseout="rh()">TOPMETR:12</a></span>;<br/>
<a NAME="E13:22_1"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  is    <a href="xxreal_2.html#M2" title="XXREAL_2:mode.2">LowerBound</a> of <font color="Maroon" title="c3">P</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">r</font> be   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xxreal_2.html#D2" onmouseover="rs('xxreal_2/D2')" onmouseout="rh()">XXREAL_2:def 2</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">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> or  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">r</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_1_3"/>
<font color="Maroon" title="c9">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</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="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">r</font></span><br/>

<a NAME="E2:22_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a></span>;<br/>
<a NAME="E3:22_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> &amp; <font color="Olive" title="b1">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" onmouseover="rs('rcomp_1/D1')" onmouseout="rh()">RCOMP_1:def 1</a></span>;<br/>
<a NAME="E4:22_1_3"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">u</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r</font> &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
;<br/>
<span class="kw">hence </span><a NAME="E5:22_1_3"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">r</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="E14:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A10</font></span>: 
<font color="Maroon" title="c3">P</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a> 
 
;<br/>
<a NAME="E15:22_1"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  is    <a href="xxreal_2.html#M2" title="XXREAL_2:mode.2">LowerBound</a> of <font color="Maroon" title="c4">Q</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">r</font> be   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xxreal_2.html#D2" onmouseover="rs('xxreal_2/D2')" onmouseout="rh()">XXREAL_2:def 2</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">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font> or  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">r</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_1_4"/>
<font color="Maroon" title="c9">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</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="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">r</font></span><br/>

<a NAME="E2:22_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c9">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a></span>;<br/>
<a NAME="E3:22_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c9">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> &amp; <font color="Olive" title="b1">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" onmouseover="rs('rcomp_1/D1')" onmouseout="rh()">RCOMP_1:def 1</a></span>;<br/>
<a NAME="E4:22_1_4"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">u</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r</font> &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
;<br/>
<span class="kw">hence </span><a NAME="E5:22_1_4"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">r</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="E16:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A11</font></span>: 
<font color="Maroon" title="c4">Q</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a> 
 
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">A0</font> = <font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c10">B0</font> = <font color="Maroon" title="c4">Q</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D13" onmouseover="rs('metric_1/D13')" onmouseout="rh()">METRIC_1:def 13</a>, <a class="ref" href="topmetr.html#T12" onmouseover="rs('topmetr/T12')" onmouseout="rh()">TOPMETR:12</a>, <a class="ref" href="topmetr.html#D6" onmouseover="rs('topmetr/D6')" onmouseout="rh()">TOPMETR:def 6</a></span>;<br/>
<a NAME="E18:22_1"/><span class="lab"><font color="Green" title="E28">A12</font></span>: 
 <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a>  is   <a href="borsuk_1.html#V1" title="BORSUK_1:attr.1">closed</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of  <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="treal_1.html#T2" target="_self" onmouseover="rs('treal_1/T2')" onmouseout="rh()">Th2</a>, <a class="ref" href="topmetr.html#T20" onmouseover="rs('topmetr/T20')" onmouseout="rh()">TOPMETR:20</a></span>;<br/>
<a NAME="E19:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A13</font></span>: 
<font color="Maroon" title="c9">A0</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="tsep_1.html#T12" onmouseover="rs('tsep_1/T12')" onmouseout="rh()">TSEP_1:12</a></span>;<br/>
<a NAME="E20:22_1"/><span class="lab"><font color="Green" title="E30">A14</font></span>: 
<font color="Maroon" title="c10">B0</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_1"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E18:22_1"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="ref" href="tsep_1.html#T12" onmouseover="rs('tsep_1/T12')" onmouseout="rh()">TSEP_1:12</a></span>;<br/>
<a NAME="E21:22_1"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> &amp; <font color="Olive" title="b1">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) </span> } </span> 
 
;<br/>
<a NAME="E22:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A15</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" onmouseover="rs('rcomp_1/D1')" onmouseout="rh()">RCOMP_1:def 1</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_1_5"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_5_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="E1:22_1_5_1"/>
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">P</font> or  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Q</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E22:22_1"><span class="lab"><font color="Green" title="E31">A15</font></span></a>, <a class="ref" href="borsuk_1.html#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:22_1_5_1_1"/><span class="lab"><font color="Green" title="E32">A16</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">P</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"> contradiction</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c11">B00</font> = <font color="Maroon" title="c10">B0</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ;<br/><span class="kw">set </span><font color="Maroon" title="c12">l</font> =  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c4">Q</font>;<br/><a NAME="E3:22_1_5_1_1"/>
 <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c4">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">m</font> =  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c4">Q</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="metric_1.html#K7" title="METRIC_1:func.7">RealSpace</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D13" onmouseover="rs('metric_1/D13')" onmouseout="rh()">METRIC_1:def 13</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">t</font> = <font color="Maroon" title="c13">m</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T12" onmouseover="rs('topmetr/T12')" onmouseout="rh()">TOPMETR:12</a>, <a class="ref" href="topmetr.html#D6" onmouseover="rs('topmetr/D6')" onmouseout="rh()">TOPMETR:def 6</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c15">W</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> &amp; <font color="Olive" title="b1">w</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c4">Q</font> ) </span> } </span> ;<br/><a NAME="E6:22_1_5_1_1"/><span class="lab"><font color="Green" title="E33">A17</font></span>: 
 <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c4">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_1_5_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:22_1_5_1_1_1"/>
 not  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c4">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</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"> contradiction</span><br/>

<a NAME="E2:22_1_5_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A18</font></span>: 
<font color="Maroon" title="c14">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">B00</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T29" onmouseover="rs('subset_1/T29')" onmouseout="rh()">SUBSET_1:29</a></span>;<br/>
<a NAME="E3:22_1_5_1_1_1"/>
<font color="Maroon" title="c11">B00</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="#E20:22_1"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="ref" href="tops_1.html#T3" onmouseover="rs('tops_1/T3')" onmouseout="rh()">TOPS_1:3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">s</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E5:22_1_5_1_1_1"/><span class="lab"><font color="Green" title="E35">A19</font></span>: 
<font color="Maroon" title="c16">s</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="E6:22_1_5_1_1_1"/><span class="lab"><font color="Green" title="E36">A20</font></span>: 
 <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c13">m</font>,<font color="Maroon" title="c16">s</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B0</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_1_5_1_1_1"><span class="lab"><font color="Green" title="E34">A18</font></span></a>, <a class="ref" href="topmetr.html#T15" onmouseover="rs('topmetr/T15')" onmouseout="rh()">TOPMETR:15</a>, <a class="ref" href="topmetr.html#D6" onmouseover="rs('topmetr/D6')" onmouseout="rh()">TOPMETR:def 6</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c17">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="E8:22_1_5_1_1_1"/><span class="lab"><font color="Green" title="E37">A21</font></span>: 
<font color="Maroon" title="c17">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font>
 <span class="kw">and </span><br/><a NAME="E9:22_1_5_1_1_1"/><span class="lab"><font color="Green" title="E38">A22</font></span>: 
<font color="Maroon" title="c17">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c4">Q</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c16">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="txt" href="#E16:22_1"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="txt" href="#E5:22_1_5_1_1_1"><span class="lab"><font color="Green" title="E35">A19</font></span></a>, <a class="ref" href="seq_4.html#D2" onmouseover="rs('seq_4/D2')" onmouseout="rh()">SEQ_4:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">r</font> = <font color="Maroon" title="c17">r</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/>
<a NAME="E11:22_1_5_1_1_1"/>
 <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c4">Q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:22_1"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="txt" href="#E8:22_1_5_1_1_1"><span class="lab"><font color="Green" title="E37">A21</font></span></a>, <a class="ref" href="seq_4.html#D2" onmouseover="rs('seq_4/D2')" onmouseout="rh()">SEQ_4:def 2</a></span>;<br/>
<a NAME="E12:22_1_5_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c4">Q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T47" onmouseover="rs('xreal_1/T47')" onmouseout="rh()">XREAL_1:47</a></span>;<br/>
<a NAME="E13:22_1_5_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A23</font></span>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c16">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c4">Q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">r</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_1_5_1_1_1"><span class="lab"><font color="Green" title="E35">A19</font></span></a>, <a class="ref" href="xreal_1.html#T24" onmouseover="rs('xreal_1/T24')" onmouseout="rh()">XREAL_1:24</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">rm</font> = <font color="Maroon" title="c18">r</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="metric_1.html#K7" title="METRIC_1:func.7">RealSpace</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D13" onmouseover="rs('metric_1/D13')" onmouseout="rh()">METRIC_1:def 13</a></span>;<br/>
<a NAME="E15:22_1_5_1_1_1"/>
<font color="Maroon" title="c18">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c4">Q</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_1_5_1_1_1"><span class="lab"><font color="Green" title="E38">A22</font></span></a>, <a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/>
<a NAME="E16:22_1_5_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c4">Q</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22_1_5_1_1_1"><span class="lab"><font color="Green" title="E39">A23</font></span></a>, <a class="ref" href="seq_2.html#T1" onmouseover="rs('seq_2/T1')" onmouseout="rh()">SEQ_2:1</a></span>;<br/>
<a NAME="E17:22_1_5_1_1_1"/><span class="kw">then </span>
 the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#K7" title="METRIC_1:func.7">RealSpace</a> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c19">rm</font>,<font color="Maroon" title="c13">m</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D12" onmouseover="rs('metric_1/D12')" onmouseout="rh()">METRIC_1:def 12</a>, <a class="ref" href="metric_1.html#D13" onmouseover="rs('metric_1/D13')" onmouseout="rh()">METRIC_1:def 13</a></span>;<br/>
<a NAME="E18:22_1_5_1_1_1"/><span class="kw">then </span>
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c13">m</font>,<font color="Maroon" title="c19">rm</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D1" onmouseover="rs('metric_1/D1')" onmouseout="rh()">METRIC_1:def 1</a></span>;<br/>
<a NAME="E19:22_1_5_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">rm</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="metric_1.html#K7" title="METRIC_1:func.7">RealSpace</a> :  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c13">m</font>,<font color="Olive" title="b1">q</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">s</font> </span> } </span> 
 
;<br/>
<a NAME="E20:22_1_5_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">rm</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="c13">m</font>,<font color="Maroon" title="c16">s</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T17" onmouseover="rs('metric_1/T17')" onmouseout="rh()">METRIC_1:17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:22_1_5_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22_1_5_1_1_1"><span class="lab"><font color="Green" title="E36">A20</font></span></a>, <a class="txt" href="#E8:22_1_5_1_1_1"><span class="lab"><font color="Green" title="E37">A21</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</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="E7:22_1_5_1_1"/><span class="kw">then </span>
 <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c4">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a></span>;<br/><a NAME="E8:22_1_5_1_1"/><span class="kw">then </span>
 <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c4">Q</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="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" onmouseover="rs('rcomp_1/D1')" onmouseout="rh()">RCOMP_1:def 1</a></span>;<br/><a NAME="E9:22_1_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A24</font></span>: 
 ex <font color="Olive" title="b1">u0</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">u0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c4">Q</font> &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">u0</font> &amp; <font color="Olive" title="b1">u0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_1_5_1_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">w</font> where <font color="Olive" title="b2">w</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">w</font> &amp; <font color="Olive" title="b2">w</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c4">Q</font> ) </span> } </span>  holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">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="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> &amp; <font color="Olive" title="b1">w</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c4">Q</font> ) </span> } </span>  implies <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> )</span><br/><span class="kw">assume </span><a NAME="E1:22_1_5_1_1_2"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> &amp; <font color="Olive" title="b1">w</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c4">Q</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="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">w0</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E3:22_1_5_1_1_2"/><span class="lab"><font color="Green" title="E35">A25</font></span>: 
<font color="Maroon" title="c17">w0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font>
 <span class="kw">and </span><br/><a NAME="E4:22_1_5_1_1_2"/><span class="lab"><font color="Green" title="E36">A26</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">w0</font>
 <span class="kw">and </span><br/><a NAME="E5:22_1_5_1_1_2"/><span class="lab"><font color="Green" title="E37">A27</font></span>: 
<font color="Maroon" title="c17">w0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c4">Q</font>
 ;<br/><a NAME="E6:22_1_5_1_1_2"/>
<font color="Maroon" title="c17">w0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_1_5_1_1"><span class="lab"><font color="Green" title="E34">A24</font></span></a>, <a class="txt" href="#E5:22_1_5_1_1_2"><span class="lab"><font color="Green" title="E37">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="E7:22_1_5_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">w0</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="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_1_5_1_1_2"><span class="lab"><font color="Green" title="E36">A26</font></span></a></span>;<br/><a NAME="E8:22_1_5_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">w0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="borsuk_1.html#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a>, <a class="ref" href="rcomp_1.html#D1" onmouseover="rs('rcomp_1/D1')" onmouseout="rh()">RCOMP_1:def 1</a></span>;<br/><a NAME="E9:22_1_5_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c17">w0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> or <font color="Maroon" title="c17">w0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E10:22_1_5_1_1_2"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:22_1"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="txt" href="#E3:22_1_5_1_1_2"><span class="lab"><font color="Green" title="E35">A25</font></span></a>, <a class="txt" href="#E5:22_1_5_1_1_2"><span class="lab"><font color="Green" title="E37">A27</font></span></a>, <a class="ref" href="seq_4.html#D2" onmouseover="rs('seq_4/D2')" onmouseout="rh()">SEQ_4:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E11:22_1_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A28</font></span>: 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> &amp; <font color="Olive" title="b1">w</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c4">Q</font> ) </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P</font>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">D</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> &amp; <font color="Olive" title="b1">w</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c4">Q</font> ) </span> } </span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:22_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="ref" href="metric_1.html#D13" onmouseover="rs('metric_1/D13')" onmouseout="rh()">METRIC_1:def 13</a>, <a class="ref" href="topmetr.html#D6" onmouseover="rs('topmetr/D6')" onmouseout="rh()">TOPMETR:def 6</a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/><a NAME="E13:22_1_5_1_1"/><span class="lab"><font color="Green" title="E36">A29</font></span>: 
 not  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E1:22_1_5_1_1"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_1_5_1_1_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  st <font color="Olive" title="b1">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c14">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">G</font> holds <br/><font color="Maroon" title="c16">D</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">G</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">G</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>; <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">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c14">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c17">G</font> implies <font color="Maroon" title="c16">D</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c17">G</font> )</span><br/><span class="kw">assume </span><a NAME="E1:22_1_5_1_1_3"/><span class="lab"><font color="Green" title="E37">A30</font></span>: 
<font color="Maroon" title="c17">G</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"> ( <font color="Maroon" title="c14">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c17">G</font> implies <font color="Maroon" title="c16">D</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c17">G</font> )</span><br/><span class="kw">assume </span><a NAME="E2:22_1_5_1_1_3"/>
<font color="Maroon" title="c14">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c17">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" title="c16">D</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c17">G</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">e</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:22_1_5_1_1_3"/><span class="lab"><font color="Green" title="E38">A31</font></span>: 
<font color="Maroon" title="c18">e</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="E5:22_1_5_1_1_3"/><span class="lab"><font color="Green" title="E39">A32</font></span>: 
 <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c13">m</font>,<font color="Maroon" title="c18">e</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1_5_1_1_3"><span class="lab"><font color="Green" title="E37">A30</font></span></a>, <a class="ref" href="topmetr.html#T15" onmouseover="rs('topmetr/T15')" onmouseout="rh()">TOPMETR:15</a>, <a class="ref" href="topmetr.html#D6" onmouseover="rs('topmetr/D6')" onmouseout="rh()">TOPMETR:def 6</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">e</font> = <font color="Maroon" title="c18">e</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">e0</font> =  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c4">Q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">e</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span>) as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">e1</font> = <font color="Maroon" title="c20">e0</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="metric_1.html#K7" title="METRIC_1:func.7">RealSpace</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D13" onmouseover="rs('metric_1/D13')" onmouseout="rh()">METRIC_1:def 13</a></span>;<br/><a NAME="E9:22_1_5_1_1_3"/><span class="lab"><font color="Green" title="E40">A33</font></span>: 
<font color="Maroon" title="c19">e</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">e</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_1_5_1_1_3"><span class="lab"><font color="Green" title="E38">A31</font></span></a>, <a class="ref" href="xreal_1.html#T68" onmouseover="rs('xreal_1/T68')" onmouseout="rh()">XREAL_1:68</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_1_5_1_1_3_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c4">Q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">e0</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">e</font></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_5_1_1_3_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:22_1_5_1_1_3_1_1"/>
( <font color="Maroon" title="c20">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c20">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c4">Q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">e</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T16" onmouseover="rs('xxreal_0/T16')" onmouseout="rh()">XXREAL_0:16</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_5_1_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:22_1_5_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E41">A34</font></span>: 
<font color="Maroon" title="c20">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c4">Q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">e0</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">e</font></span><br/><div class="add"><a NAME="E2:22_1_5_1_1_3_1_1_1"/><span class="kw">then </span>
 <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c4">Q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">e</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T50" onmouseover="rs('xreal_1/T50')" onmouseout="rh()">XREAL_1:50</a>, <a class="ref" href="xxreal_0.html#T25" onmouseover="rs('xxreal_0/T25')" onmouseout="rh()">XXREAL_0:25</a></span>;<br/><a NAME="E3:22_1_5_1_1_3_1_1_1"/><span class="kw">then </span>
 <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c4">Q</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_1_5_1_1_3"><span class="lab"><font color="Green" title="E40">A33</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E4:22_1_5_1_1_3_1_1_1"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c4">Q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">e0</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_1_5_1_1"><span class="lab"><font color="Green" title="E34">A24</font></span></a>, <a class="txt" href="#E1:22_1_5_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E41">A34</font></span></a>, <a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_5_1_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:22_1_5_1_1_3_1_1_2"/>
<font color="Maroon" title="c20">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c4">Q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">e</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</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"> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c4">Q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">e0</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">e</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:22_1_5_1_1_3_1_1_2"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c4">Q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">e0</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_1_5_1_1_3"><span class="lab"><font color="Green" title="E38">A31</font></span></a>, <a class="txt" href="#E9:22_1_5_1_1_3"><span class="lab"><font color="Green" title="E40">A33</font></span></a>, <a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E11:22_1_5_1_1_3"/><span class="kw">then </span>
 the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#K7" title="METRIC_1:func.7">RealSpace</a> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c13">m</font>,<font color="Maroon" title="c21">e1</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">e</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D12" onmouseover="rs('metric_1/D12')" onmouseout="rh()">METRIC_1:def 12</a>, <a class="ref" href="metric_1.html#D13" onmouseover="rs('metric_1/D13')" onmouseout="rh()">METRIC_1:def 13</a></span>;<br/><a NAME="E12:22_1_5_1_1_3"/><span class="kw">then </span>
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c13">m</font>,<font color="Maroon" title="c21">e1</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">e</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D1" onmouseover="rs('metric_1/D1')" onmouseout="rh()">METRIC_1:def 1</a></span>;<br/><a NAME="E13:22_1_5_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c21">e1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="metric_1.html#K7" title="METRIC_1:func.7">RealSpace</a> :  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c13">m</font>,<font color="Olive" title="b1">z</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">e</font> </span> } </span> 
 ;<br/><a NAME="E14:22_1_5_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A35</font></span>: 
<font color="Maroon" title="c21">e1</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="c13">m</font>,<font color="Maroon" title="c19">e</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T17" onmouseover="rs('metric_1/T17')" onmouseout="rh()">METRIC_1:17</a></span>;<br/><a NAME="E15:22_1_5_1_1_3"/>
( <font color="Maroon" title="c20">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c20">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c4">Q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">e</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#T16" onmouseover="rs('xxreal_0/T16')" onmouseout="rh()">XXREAL_0:16</a></span>;<br/><a NAME="E16:22_1_5_1_1_3"/><span class="kw">then </span>
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">e0</font> &amp; <font color="Maroon" title="c20">e0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c4">Q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22_1_5_1_1"><span class="lab"><font color="Green" title="E36">A29</font></span></a>, <a class="txt" href="#E6:22_1_5_1_1"><span class="lab"><font color="Green" title="E33">A17</font></span></a>, <a class="txt" href="#E9:22_1_5_1_1"><span class="lab"><font color="Green" title="E34">A24</font></span></a>, <a class="txt" href="#E4:22_1_5_1_1_3"><span class="lab"><font color="Green" title="E38">A31</font></span></a>, <a class="ref" href="xreal_1.html#T44" onmouseover="rs('xreal_1/T44')" onmouseout="rh()">XREAL_1:44</a>, <a class="ref" href="xreal_1.html#T139" onmouseover="rs('xreal_1/T139')" onmouseout="rh()">XREAL_1:139</a>, <a class="ref" href="xxreal_0.html#T25" onmouseover="rs('xxreal_0/T25')" onmouseout="rh()">XXREAL_0:25</a></span>;<br/><a NAME="E17:22_1_5_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c20">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> &amp; <font color="Olive" title="b1">w</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c4">Q</font> ) </span> } </span> 
 ;<br/><span class="kw">hence </span><a NAME="E18:22_1_5_1_1_3"/>
<font color="Maroon" title="c16">D</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c17">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_1_5_1_1_3"><span class="lab"><font color="Green" title="E39">A32</font></span></a>, <a class="txt" href="#E14:22_1_5_1_1_3"><span class="lab"><font color="Green" title="E41">A35</font></span></a>, <a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E15:22_1_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A36</font></span>: 
<font color="Maroon" title="c14">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c16">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T24" onmouseover="rs('pre_topc/T24')" onmouseout="rh()">PRE_TOPC:24</a></span>;<br/><a NAME="E16:22_1_5_1_1"/><span class="lab"><font color="Green" title="E38">A37</font></span>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c9">A0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:22_1"><span class="lab"><font color="Green" title="E29">A13</font></span></a>, <a class="ref" href="pre_topc.html#T22" onmouseover="rs('pre_topc/T22')" onmouseout="rh()">PRE_TOPC:22</a></span>;<br/><a NAME="E17:22_1_5_1_1"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c16">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c9">A0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22_1_5_1_1"><span class="lab"><font color="Green" title="E35">A28</font></span></a>, <a class="ref" href="pre_topc.html#T19" onmouseover="rs('pre_topc/T19')" onmouseout="rh()">PRE_TOPC:19</a></span>;<br/><span class="kw">hence </span><a NAME="E18:22_1_5_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E6:22_1_5_1_1"><span class="lab"><font color="Green" title="E33">A17</font></span></a>, <a class="txt" href="#E15:22_1_5_1_1"><span class="lab"><font color="Green" title="E37">A36</font></span></a>, <a class="txt" href="#E16:22_1_5_1_1"><span class="lab"><font color="Green" title="E38">A37</font></span></a>, <a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:22_1_5_1_2"/><span class="lab"><font color="Green" title="E32">A38</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Q</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"> contradiction</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c11">A00</font> = <font color="Maroon" title="c9">A0</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ;<br/><span class="kw">set </span><font color="Maroon" title="c12">l</font> =  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</font>;<br/><a NAME="E3:22_1_5_1_2"/>
 <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">m</font> =  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="metric_1.html#K7" title="METRIC_1:func.7">RealSpace</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D13" onmouseover="rs('metric_1/D13')" onmouseout="rh()">METRIC_1:def 13</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">t</font> = <font color="Maroon" title="c13">m</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T12" onmouseover="rs('topmetr/T12')" onmouseout="rh()">TOPMETR:12</a>, <a class="ref" href="topmetr.html#D6" onmouseover="rs('topmetr/D6')" onmouseout="rh()">TOPMETR:def 6</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c15">W</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> &amp; <font color="Olive" title="b1">w</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</font> ) </span> } </span> ;<br/><a NAME="E6:22_1_5_1_2"/><span class="lab"><font color="Green" title="E33">A39</font></span>: 
 <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_1_5_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:22_1_5_1_2_1"/>
 not  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</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"> contradiction</span><br/>

<a NAME="E2:22_1_5_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A40</font></span>: 
<font color="Maroon" title="c14">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">A00</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T29" onmouseover="rs('subset_1/T29')" onmouseout="rh()">SUBSET_1:29</a></span>;<br/>
<a NAME="E3:22_1_5_1_2_1"/>
<font color="Maroon" title="c11">A00</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="#E19:22_1"><span class="lab"><font color="Green" title="E29">A13</font></span></a>, <a class="ref" href="tops_1.html#T3" onmouseover="rs('tops_1/T3')" onmouseout="rh()">TOPS_1:3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">s</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E5:22_1_5_1_2_1"/><span class="lab"><font color="Green" title="E35">A41</font></span>: 
<font color="Maroon" title="c16">s</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="E6:22_1_5_1_2_1"/><span class="lab"><font color="Green" title="E36">A42</font></span>: 
 <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c13">m</font>,<font color="Maroon" title="c16">s</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">A0</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_1_5_1_2_1"><span class="lab"><font color="Green" title="E34">A40</font></span></a>, <a class="ref" href="topmetr.html#T15" onmouseover="rs('topmetr/T15')" onmouseout="rh()">TOPMETR:15</a>, <a class="ref" href="topmetr.html#D6" onmouseover="rs('topmetr/D6')" onmouseout="rh()">TOPMETR:def 6</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c17">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="E8:22_1_5_1_2_1"/><span class="lab"><font color="Green" title="E37">A43</font></span>: 
<font color="Maroon" title="c17">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E9:22_1_5_1_2_1"/><span class="lab"><font color="Green" title="E38">A44</font></span>: 
<font color="Maroon" title="c17">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c16">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E14:22_1"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="txt" href="#E5:22_1_5_1_2_1"><span class="lab"><font color="Green" title="E35">A41</font></span></a>, <a class="ref" href="seq_4.html#D2" onmouseover="rs('seq_4/D2')" onmouseout="rh()">SEQ_4:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">r</font> = <font color="Maroon" title="c17">r</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/>
<a NAME="E11:22_1_5_1_2_1"/>
 <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:22_1"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="txt" href="#E8:22_1_5_1_2_1"><span class="lab"><font color="Green" title="E37">A43</font></span></a>, <a class="ref" href="seq_4.html#D2" onmouseover="rs('seq_4/D2')" onmouseout="rh()">SEQ_4:def 2</a></span>;<br/>
<a NAME="E12:22_1_5_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T47" onmouseover="rs('xreal_1/T47')" onmouseout="rh()">XREAL_1:47</a></span>;<br/>
<a NAME="E13:22_1_5_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A45</font></span>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c16">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">r</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_1_5_1_2_1"><span class="lab"><font color="Green" title="E35">A41</font></span></a>, <a class="ref" href="xreal_1.html#T24" onmouseover="rs('xreal_1/T24')" onmouseout="rh()">XREAL_1:24</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">rm</font> = <font color="Maroon" title="c18">r</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="metric_1.html#K7" title="METRIC_1:func.7">RealSpace</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D13" onmouseover="rs('metric_1/D13')" onmouseout="rh()">METRIC_1:def 13</a></span>;<br/>
<a NAME="E15:22_1_5_1_2_1"/><span class="lab"><font color="Green" title="E40">A46</font></span>: 
<a href="metric_1.html#K6" title="METRIC_1:func.6">real_dist</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c18">r</font>,<span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c19">rm</font>,<font color="Maroon" title="c13">m</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D1" onmouseover="rs('metric_1/D1')" onmouseout="rh()">METRIC_1:def 1</a>, <a class="ref" href="metric_1.html#D13" onmouseover="rs('metric_1/D13')" onmouseout="rh()">METRIC_1:def 13</a></span>;<br/>
<a NAME="E16:22_1_5_1_2_1"/>
<font color="Maroon" title="c18">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_1_5_1_2_1"><span class="lab"><font color="Green" title="E38">A44</font></span></a>, <a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/>
<a NAME="E17:22_1_5_1_2_1"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22_1_5_1_2_1"><span class="lab"><font color="Green" title="E39">A45</font></span></a>, <a class="ref" href="seq_2.html#T1" onmouseover="rs('seq_2/T1')" onmouseout="rh()">SEQ_2:1</a></span>;<br/>
<a NAME="E18:22_1_5_1_2_1"/><span class="kw">then </span>
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c19">rm</font>,<font color="Maroon" title="c13">m</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D12" onmouseover="rs('metric_1/D12')" onmouseout="rh()">METRIC_1:def 12</a>, <a class="txt" href="#E15:22_1_5_1_2_1"><span class="lab"><font color="Green" title="E40">A46</font></span></a></span>;<br/>
<a NAME="E19:22_1_5_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">rm</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="metric_1.html#K7" title="METRIC_1:func.7">RealSpace</a> :  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c13">m</font>,<font color="Olive" title="b1">q</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">s</font> </span> } </span> 
 
;<br/>
<a NAME="E20:22_1_5_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">rm</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="c13">m</font>,<font color="Maroon" title="c16">s</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T17" onmouseover="rs('metric_1/T17')" onmouseout="rh()">METRIC_1:17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:22_1_5_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22_1_5_1_2_1"><span class="lab"><font color="Green" title="E36">A42</font></span></a>, <a class="txt" href="#E8:22_1_5_1_2_1"><span class="lab"><font color="Green" title="E37">A43</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</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="E7:22_1_5_1_2"/><span class="kw">then </span>
 <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a></span>;<br/><a NAME="E8:22_1_5_1_2"/><span class="kw">then </span>
 <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</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="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" onmouseover="rs('rcomp_1/D1')" onmouseout="rh()">RCOMP_1:def 1</a></span>;<br/><a NAME="E9:22_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A47</font></span>: 
 ex <font color="Olive" title="b1">u0</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">u0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</font> &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">u0</font> &amp; <font color="Olive" title="b1">u0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_1_5_1_2_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">w</font> where <font color="Olive" title="b2">w</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">w</font> &amp; <font color="Olive" title="b2">w</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</font> ) </span> } </span>  holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">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="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> &amp; <font color="Olive" title="b1">w</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</font> ) </span> } </span>  implies <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font> )</span><br/><span class="kw">assume </span><a NAME="E1:22_1_5_1_2_2"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> &amp; <font color="Olive" title="b1">w</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</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="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">w0</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E3:22_1_5_1_2_2"/><span class="lab"><font color="Green" title="E35">A48</font></span>: 
<font color="Maroon" title="c17">w0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font>
 <span class="kw">and </span><br/><a NAME="E4:22_1_5_1_2_2"/><span class="lab"><font color="Green" title="E36">A49</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">w0</font>
 <span class="kw">and </span><br/><a NAME="E5:22_1_5_1_2_2"/><span class="lab"><font color="Green" title="E37">A50</font></span>: 
<font color="Maroon" title="c17">w0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</font>
 ;<br/><a NAME="E6:22_1_5_1_2_2"/>
<font color="Maroon" title="c17">w0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_1_5_1_2"><span class="lab"><font color="Green" title="E34">A47</font></span></a>, <a class="txt" href="#E5:22_1_5_1_2_2"><span class="lab"><font color="Green" title="E37">A50</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="E7:22_1_5_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">w0</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="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_1_5_1_2_2"><span class="lab"><font color="Green" title="E36">A49</font></span></a></span>;<br/><a NAME="E8:22_1_5_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">w0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="borsuk_1.html#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a>, <a class="ref" href="rcomp_1.html#D1" onmouseover="rs('rcomp_1/D1')" onmouseout="rh()">RCOMP_1:def 1</a></span>;<br/><a NAME="E9:22_1_5_1_2_2"/><span class="kw">then </span>
( <font color="Maroon" title="c17">w0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> or <font color="Maroon" title="c17">w0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E10:22_1_5_1_2_2"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:22_1"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="txt" href="#E3:22_1_5_1_2_2"><span class="lab"><font color="Green" title="E35">A48</font></span></a>, <a class="txt" href="#E5:22_1_5_1_2_2"><span class="lab"><font color="Green" title="E37">A50</font></span></a>, <a class="ref" href="seq_4.html#D2" onmouseover="rs('seq_4/D2')" onmouseout="rh()">SEQ_4:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E11:22_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A51</font></span>: 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> &amp; <font color="Olive" title="b1">w</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</font> ) </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Q</font>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">D</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> &amp; <font color="Olive" title="b1">w</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</font> ) </span> } </span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:22_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="ref" href="metric_1.html#D13" onmouseover="rs('metric_1/D13')" onmouseout="rh()">METRIC_1:def 13</a>, <a class="ref" href="topmetr.html#D6" onmouseover="rs('topmetr/D6')" onmouseout="rh()">TOPMETR:def 6</a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/><a NAME="E13:22_1_5_1_2"/><span class="lab"><font color="Green" title="E36">A52</font></span>: 
 not  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E1:22_1_5_1_2"><span class="lab"><font color="Green" title="E32">A38</font></span></a>, <a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_1_5_1_2_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  st <font color="Olive" title="b1">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c14">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">G</font> holds <br/><font color="Maroon" title="c16">D</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">G</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">G</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>; <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">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c14">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c17">G</font> implies <font color="Maroon" title="c16">D</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c17">G</font> )</span><br/><span class="kw">assume </span><a NAME="E1:22_1_5_1_2_3"/><span class="lab"><font color="Green" title="E37">A53</font></span>: 
<font color="Maroon" title="c17">G</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"> ( <font color="Maroon" title="c14">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c17">G</font> implies <font color="Maroon" title="c16">D</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c17">G</font> )</span><br/><span class="kw">assume </span><a NAME="E2:22_1_5_1_2_3"/>
<font color="Maroon" title="c14">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c17">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" title="c16">D</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c17">G</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">e</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:22_1_5_1_2_3"/><span class="lab"><font color="Green" title="E38">A54</font></span>: 
<font color="Maroon" title="c18">e</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="E5:22_1_5_1_2_3"/><span class="lab"><font color="Green" title="E39">A55</font></span>: 
 <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c13">m</font>,<font color="Maroon" title="c18">e</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1_5_1_2_3"><span class="lab"><font color="Green" title="E37">A53</font></span></a>, <a class="ref" href="topmetr.html#T15" onmouseover="rs('topmetr/T15')" onmouseout="rh()">TOPMETR:15</a>, <a class="ref" href="topmetr.html#D6" onmouseover="rs('topmetr/D6')" onmouseout="rh()">TOPMETR:def 6</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">e</font> = <font color="Maroon" title="c18">e</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">e0</font> =  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">e</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span>) as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">e1</font> = <font color="Maroon" title="c20">e0</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="metric_1.html#K7" title="METRIC_1:func.7">RealSpace</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D13" onmouseover="rs('metric_1/D13')" onmouseout="rh()">METRIC_1:def 13</a></span>;<br/><a NAME="E9:22_1_5_1_2_3"/><span class="lab"><font color="Green" title="E40">A56</font></span>: 
<font color="Maroon" title="c19">e</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">e</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_1_5_1_2_3"><span class="lab"><font color="Green" title="E38">A54</font></span></a>, <a class="ref" href="xreal_1.html#T68" onmouseover="rs('xreal_1/T68')" onmouseout="rh()">XREAL_1:68</a></span>;<br/><a NAME="E10:22_1_5_1_2_3"/><span class="lab"><font color="Green" title="E41">A57</font></span>: 
<a href="metric_1.html#K6" title="METRIC_1:func.6">real_dist</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</font></span>)</span>,<font color="Maroon" title="c20">e0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c13">m</font>,<font color="Maroon" title="c21">e1</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D1" onmouseover="rs('metric_1/D1')" onmouseout="rh()">METRIC_1:def 1</a>, <a class="ref" href="metric_1.html#D13" onmouseover="rs('metric_1/D13')" onmouseout="rh()">METRIC_1:def 13</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_1_5_1_2_3_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">e0</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">e</font></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_5_1_2_3_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:22_1_5_1_2_3_1_1"/>
( <font color="Maroon" title="c20">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c20">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">e</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T16" onmouseover="rs('xxreal_0/T16')" onmouseout="rh()">XXREAL_0:16</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_5_1_2_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:22_1_5_1_2_3_1_1_1"/><span class="lab"><font color="Green" title="E42">A58</font></span>: 
<font color="Maroon" title="c20">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">e0</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">e</font></span><br/><div class="add"><a NAME="E2:22_1_5_1_2_3_1_1_1"/><span class="kw">then </span>
 <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">e</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T50" onmouseover="rs('xreal_1/T50')" onmouseout="rh()">XREAL_1:50</a>, <a class="ref" href="xxreal_0.html#T25" onmouseover="rs('xxreal_0/T25')" onmouseout="rh()">XXREAL_0:25</a></span>;<br/><a NAME="E3:22_1_5_1_2_3_1_1_1"/><span class="kw">then </span>
 <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_1_5_1_2_3"><span class="lab"><font color="Green" title="E40">A56</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E4:22_1_5_1_2_3_1_1_1"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">e0</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_1_5_1_2"><span class="lab"><font color="Green" title="E34">A47</font></span></a>, <a class="txt" href="#E1:22_1_5_1_2_3_1_1_1"><span class="lab"><font color="Green" title="E42">A58</font></span></a>, <a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_5_1_2_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:22_1_5_1_2_3_1_1_2"/>
<font color="Maroon" title="c20">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">e</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</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"> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">e0</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">e</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:22_1_5_1_2_3_1_1_2"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">e0</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_1_5_1_2_3"><span class="lab"><font color="Green" title="E38">A54</font></span></a>, <a class="txt" href="#E9:22_1_5_1_2_3"><span class="lab"><font color="Green" title="E40">A56</font></span></a>, <a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E12:22_1_5_1_2_3"/><span class="kw">then </span>
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c13">m</font>,<font color="Maroon" title="c21">e1</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">e</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D12" onmouseover="rs('metric_1/D12')" onmouseout="rh()">METRIC_1:def 12</a>, <a class="txt" href="#E10:22_1_5_1_2_3"><span class="lab"><font color="Green" title="E41">A57</font></span></a></span>;<br/><a NAME="E13:22_1_5_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c21">e1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="metric_1.html#K7" title="METRIC_1:func.7">RealSpace</a> :  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c13">m</font>,<font color="Olive" title="b1">z</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">e</font> </span> } </span> 
 ;<br/><a NAME="E14:22_1_5_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A59</font></span>: 
<font color="Maroon" title="c21">e1</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="c13">m</font>,<font color="Maroon" title="c19">e</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T17" onmouseover="rs('metric_1/T17')" onmouseout="rh()">METRIC_1:17</a></span>;<br/><a NAME="E15:22_1_5_1_2_3"/>
( <font color="Maroon" title="c20">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c20">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">e</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#T16" onmouseover="rs('xxreal_0/T16')" onmouseout="rh()">XXREAL_0:16</a></span>;<br/><a NAME="E16:22_1_5_1_2_3"/><span class="kw">then </span>
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">e0</font> &amp; <font color="Maroon" title="c20">e0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22_1_5_1_2"><span class="lab"><font color="Green" title="E36">A52</font></span></a>, <a class="txt" href="#E6:22_1_5_1_2"><span class="lab"><font color="Green" title="E33">A39</font></span></a>, <a class="txt" href="#E9:22_1_5_1_2"><span class="lab"><font color="Green" title="E34">A47</font></span></a>, <a class="txt" href="#E4:22_1_5_1_2_3"><span class="lab"><font color="Green" title="E38">A54</font></span></a>, <a class="ref" href="xreal_1.html#T44" onmouseover="rs('xreal_1/T44')" onmouseout="rh()">XREAL_1:44</a>, <a class="ref" href="xreal_1.html#T139" onmouseover="rs('xreal_1/T139')" onmouseout="rh()">XREAL_1:139</a>, <a class="ref" href="xxreal_0.html#T25" onmouseover="rs('xxreal_0/T25')" onmouseout="rh()">XXREAL_0:25</a></span>;<br/><a NAME="E17:22_1_5_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c20">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> &amp; <font color="Olive" title="b1">w</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">P</font> ) </span> } </span> 
 ;<br/><span class="kw">hence </span><a NAME="E18:22_1_5_1_2_3"/>
<font color="Maroon" title="c16">D</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c17">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_1_5_1_2_3"><span class="lab"><font color="Green" title="E39">A55</font></span></a>, <a class="txt" href="#E14:22_1_5_1_2_3"><span class="lab"><font color="Green" title="E42">A59</font></span></a>, <a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E15:22_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A60</font></span>: 
<font color="Maroon" title="c14">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c16">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T24" onmouseover="rs('pre_topc/T24')" onmouseout="rh()">PRE_TOPC:24</a></span>;<br/><a NAME="E16:22_1_5_1_2"/><span class="lab"><font color="Green" title="E38">A61</font></span>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c10">B0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:22_1"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="ref" href="pre_topc.html#T22" onmouseover="rs('pre_topc/T22')" onmouseout="rh()">PRE_TOPC:22</a></span>;<br/><a NAME="E17:22_1_5_1_2"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c16">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c10">B0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22_1_5_1_2"><span class="lab"><font color="Green" title="E35">A51</font></span></a>, <a class="ref" href="pre_topc.html#T19" onmouseover="rs('pre_topc/T19')" onmouseout="rh()">PRE_TOPC:19</a></span>;<br/><span class="kw">hence </span><a NAME="E18:22_1_5_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E6:22_1_5_1_2"><span class="lab"><font color="Green" title="E33">A39</font></span></a>, <a class="txt" href="#E15:22_1_5_1_2"><span class="lab"><font color="Green" title="E37">A60</font></span></a>, <a class="txt" href="#E16:22_1_5_1_2"><span class="lab"><font color="Green" title="E38">A61</font></span></a>, <a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E24:22_1"/>
contradiction
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:22"/>
 <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a>  is  <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_1.html#T10" onmouseover="rs('connsp_1/T10')" onmouseout="rh()">CONNSP_1:10</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>
