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

<span class="kw">let </span><font color="Maroon" title="c1">P7</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"> ( <font color="Maroon" title="c1">P7</font> <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 <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> implies ( <font color="Maroon" title="c1">P7</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c1">P7</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:39"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
<font color="Maroon" title="c1">P7</font> <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 <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">P7</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c1">P7</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  )</span><br/>

<a NAME="E2:39"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <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="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/>
<a NAME="E3:39"/><span class="lab"><font color="Green" title="E26">A3</font></span>: 
 not 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<a NAME="E4:39"/><span class="lab"><font color="Green" title="E27">A4</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P7</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P7</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E5:39"/>
 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 <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P7</font></span>)</span>  st  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P7</font></span>)</span> <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> <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P7</font></span>)</span> &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> <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P7</font></span>)</span> &amp; <font color="Olive" title="b1">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive" title="b2">B</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</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="39_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P7</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"> (  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P7</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c2">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> <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P7</font></span>)</span> &amp; <font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P7</font></span>)</span> &amp; <font color="Maroon" title="c2">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c3">B</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">B</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:39_1"/><span class="lab"><font color="Green" title="E28">A5</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P7</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font>
 <span class="kw">and </span><br/><a NAME="E2:39_1"/><span class="lab"><font color="Green" title="E29">A6</font></span>: 
<font color="Maroon" title="c2">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> <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P7</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:39_1"/><span class="lab"><font color="Green" title="E30">A7</font></span>: 
<font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P7</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:39_1"/><span class="lab"><font color="Green" title="E31">A8</font></span>: 
<font color="Maroon" title="c2">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E5:39_1"/><span class="lab"><font color="Green" title="E32">A9</font></span>: 
<font color="Maroon" title="c3">B</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="c2">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">B</font></span><br/>

<span class="kw">assume </span><a NAME="E6:39_1"/><span class="lab"><font color="Green" title="E33">A10</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">B</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E7:39_1"/><span class="lab"><font color="Green" title="E34">A11</font></span>: 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T86" onmouseover="rs('xxreal_1/T86')" onmouseout="rh()">XXREAL_1:86</a></span>;<br/>
<span class="lab"><font color="Green" title="E35">A12</font></span>: <font color="Maroon" title="c1">P7</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E24">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="xxreal_1.html#T128" onmouseover="rs('xxreal_1/T128')" onmouseout="rh()">XXREAL_1:128</a></span>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T40" onmouseover="rs('xboole_1/T40')" onmouseout="rh()">XBOOLE_1:40</a></span>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:39_1"><span class="lab"><font color="Green" title="E34">A11</font></span></a>, <a class="ref" href="xboole_1.html#T83" onmouseover="rs('xboole_1/T83')" onmouseout="rh()">XBOOLE_1:83</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">D1</font> = <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> 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="topmetr.html#T17" onmouseover="rs('topmetr/T17')" onmouseout="rh()">TOPMETR:17</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">P1</font> = <font color="Maroon" title="c1">P7</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="txt" href="#E8:39_1"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="ref" href="topmetr.html#T17" onmouseover="rs('topmetr/T17')" onmouseout="rh()">TOPMETR:17</a></span>;<br/>
<a NAME="E11:39_1"/>
 <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T19" onmouseover="rs('topmetr/T19')" onmouseout="rh()">TOPMETR:19</a>, <a class="ref" href="topmetr.html#T20" onmouseover="rs('topmetr/T20')" onmouseout="rh()">TOPMETR:20</a></span>;<br/>
<a NAME="E12:39_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A13</font></span>: 
<a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">P1</font>
 
<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="pre_topc.html#T7" onmouseover="rs('pre_topc/T7')" onmouseout="rh()">PRE_TOPC:7</a></span>;<br/>
<a NAME="E13:39_1"/>
<font color="Maroon" title="c5">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">r1</font> where <font color="Olive" title="b1">r1</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r1</font> &amp; <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:39_1"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="ref" href="rcomp_1.html#D2" onmouseover="rs('rcomp_1/D2')" onmouseout="rh()">RCOMP_1:def 2</a></span>;<br/>
<a NAME="E14:39_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">P1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2b.html#T26" onmouseover="rs('jordan2b/T26')" onmouseout="rh()">JORDAN2B:26</a></span>;<br/>
<a NAME="E15:39_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A14</font></span>: 
<a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P7</font> is   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</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="txt" href="#E1:39"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E2:39"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E12:39_1"><span class="lab"><font color="Green" title="E36">A13</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="tsep_1.html#T16" onmouseover="rs('tsep_1/T16')" onmouseout="rh()">TSEP_1:16</a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">P</font> = <font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c7">Q</font> = <font color="Maroon" title="c3">B</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E2:39_1"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="txt" href="#E3:39_1"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="txt" href="#E8:39_1"><span class="lab"><font color="Green" title="E35">A12</font></span></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">reconsider </span><font color="Maroon" title="c8">A0</font> = <font color="Maroon" title="c6">P</font>, <font color="Maroon" title="c9">B0</font> = <font color="Maroon" title="c7">Q</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <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="E18:39_1"/><span class="lab"><font color="Green" title="E38">A15</font></span>: 
<font color="Maroon" title="c8">A0</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="txt" href="#E15:39_1"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="ref" href="tsep_1.html#T17" onmouseover="rs('tsep_1/T17')" onmouseout="rh()">TSEP_1:17</a></span>;<br/>
<a NAME="E19:39_1"/><span class="lab"><font color="Green" title="E39">A16</font></span>: 
<font color="Maroon" title="c9">B0</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="#E5:39_1"><span class="lab"><font color="Green" title="E32">A9</font></span></a>, <a class="txt" href="#E15:39_1"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="ref" href="tsep_1.html#T17" onmouseover="rs('tsep_1/T17')" onmouseout="rh()">TSEP_1:17</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">xp</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">P</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">xp</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">P</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<a NAME="E21:39_1"/><span class="lab"><font color="Green" title="E40">A17</font></span>: 
<font color="Maroon" title="c11">xp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 
;<br/>
<a NAME="E22:39_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="c6">P</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="39_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">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="c12">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">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="c12">r</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:39_1_2"/>
<font color="Maroon" title="c12">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">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="c12">r</font></span><br/>

<a NAME="E2:39_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E8:39_1"><span class="lab"><font color="Green" title="E35">A12</font></span></a></span>;<br/>
<a NAME="E3:39_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">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#NR3" title="XXREAL_0:NR.3">&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> 1 ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D2" onmouseover="rs('rcomp_1/D2')" onmouseout="rh()">RCOMP_1:def 2</a></span>;<br/>
<a NAME="E4:39_1_2"/><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="c12">r</font> &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 
;<br/>
<span class="kw">hence </span><a NAME="E5:39_1_2"/>
 <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="c12">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="E23:39_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A18</font></span>: 
<font color="Maroon" title="c6">P</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a> 
 
;<br/>
<a NAME="E24:39_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="c7">Q</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="39_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">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="c12">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">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="c12">r</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:39_1_3"/>
<font color="Maroon" title="c12">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">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="c12">r</font></span><br/>

<a NAME="E2:39_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c12">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E8:39_1"><span class="lab"><font color="Green" title="E35">A12</font></span></a></span>;<br/>
<a NAME="E3:39_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c12">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#NR3" title="XXREAL_0:NR.3">&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> 1 ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D2" onmouseover="rs('rcomp_1/D2')" onmouseout="rh()">RCOMP_1:def 2</a></span>;<br/>
<a NAME="E4:39_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="c12">r</font> &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 
;<br/>
<span class="kw">hence </span><a NAME="E5:39_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="c12">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="E25:39_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A19</font></span>: 
<font color="Maroon" title="c7">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="c12">l</font> =  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c7">Q</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="c13">m</font> = <font color="Maroon" title="c12">l</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/>
<a NAME="E29:39_1"/><span class="lab"><font color="Green" title="E43">A20</font></span>: 
 not <font color="Maroon" title="c12">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">Q</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="39_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:39_1_4"/>
<font color="Maroon" title="c12">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">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/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">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="E3:39_1_4"/><span class="lab"><font color="Green" title="E44">A21</font></span>: 
<font color="Maroon" title="c15">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="E4:39_1_4"/><span class="lab"><font color="Green" title="E45">A22</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="c15">s</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">B0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:39_1"><span class="lab"><font color="Green" title="E39">A16</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="c16">s</font> = <font color="Maroon" title="c15">s</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="c17">s2</font> = <font color="Maroon" title="c12">l</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">s</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</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="c18">e1</font> = <font color="Maroon" title="c17">s2</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="E8:39_1_4"/>
<font color="Maroon" title="c16">s</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <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="#E3:39_1_4"><span class="lab"><font color="Green" title="E44">A21</font></span></a>, <a class="ref" href="xreal_1.html#T216" onmouseover="rs('xreal_1/T216')" onmouseout="rh()">XREAL_1:216</a></span>;<br/>
<a NAME="E9:39_1_4"/><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="c12">l</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">s</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></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="#E3:39_1_4"><span class="lab"><font color="Green" title="E44">A21</font></span></a>, <a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>;<br/>
<a NAME="E10:39_1_4"/><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="c18">e1</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="E11:39_1_4"/><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="c18">e1</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="E12:39_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c18">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="c16">s</font> </span> } </span> 
 
;<br/>
<a NAME="E13:39_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c12">l</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">s</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <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/>
<a NAME="E14:39_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c12">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">l</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">s</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:39_1"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="txt" href="#E4:39_1_4"><span class="lab"><font color="Green" title="E45">A22</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="E15:39_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c12">l</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">s</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/>
<a NAME="E16:39_1_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">s</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">l</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c12">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:39_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_1_4"><span class="lab"><font color="Green" title="E44">A21</font></span></a>, <a class="ref" href="xreal_1.html#T139" onmouseover="rs('xreal_1/T139')" onmouseout="rh()">XREAL_1:139</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="lab"><font color="Green" title="E44">A23</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="39_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">  not <font color="Maroon" title="c12">l</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></a><div class="add"><span class="kw">assume </span><a NAME="E1:39_1_5"/><span class="lab"><font color="Green" title="E45">A24</font></span>: 
<font color="Maroon" title="c12">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:39_1_5"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">xp</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E8:39_1"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="txt" href="#E21:39_1"><span class="lab"><font color="Green" title="E40">A17</font></span></a>, <a class="ref" href="xxreal_1.html#T4" onmouseover="rs('xxreal_1/T4')" onmouseout="rh()">XXREAL_1:4</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">r5</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:39_1_5"/><span class="lab"><font color="Green" title="E46">A25</font></span>: 
<font color="Maroon" title="c15">r5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Q</font>
 <span class="kw">and </span><br/><a NAME="E5:39_1_5"/><span class="lab"><font color="Green" title="E47">A26</font></span>: 
<font color="Maroon" title="c15">r5</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">l</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xp</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">l</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:39_1"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="txt" href="#E1:39_1_5"><span class="lab"><font color="Green" title="E45">A24</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="c16">r5</font> = <font color="Maroon" title="c15">r5</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/><a NAME="E7:39_1_5"/><span class="lab"><font color="Green" title="E48">A27</font></span>: 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">s5</font> where <font color="Olive" title="b1">s5</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : ( <font color="Olive" title="b1">s5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c16">r5</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s5</font> ) </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">P</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="39_1_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">s5</font> where <font color="Olive" title="b1">s5</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : ( <font color="Olive" title="b1">s5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c16">r5</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s5</font> ) </span> } </span>  or <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:39_1_5_1"/>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">s5</font> where <font color="Olive" title="b1">s5</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : ( <font color="Olive" title="b1">s5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c16">r5</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s5</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="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font></span><br/>

<a NAME="E2:39_1_5_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">s5</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">s5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y</font> &amp; <font color="Olive" title="b1">s5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c16">r5</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s5</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:39_1_5_1"/>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">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"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">P5</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">s5</font> where <font color="Olive" title="b1">s5</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : ( <font color="Olive" title="b1">s5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c16">r5</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s5</font> ) </span> } </span>  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="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c18">PP5</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">s5</font> where <font color="Olive" title="b1">s5</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : ( <font color="Olive" title="b1">s5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c16">r5</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s5</font> ) </span> } </span> ;<br/><a NAME="E9:39_1_5"/><span class="lab"><font color="Green" title="E49">A28</font></span>: 
<font color="Maroon" title="c11">xp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">P5</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39_1_5"><span class="lab"><font color="Green" title="E47">A26</font></span></a></span>;<br/><a NAME="E10:39_1_5"/><span class="lab"><font color="Green" title="E50">A29</font></span>: 
<font color="Maroon" title="c17">P5</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:39_1"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="txt" href="#E7:39_1_5"><span class="lab"><font color="Green" title="E48">A27</font></span></a>, <a class="ref" href="xxreal_2.html#T44" onmouseover="rs('xxreal_2/T44')" onmouseout="rh()">XXREAL_2:44</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">l5</font> =  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c17">P5</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">m5</font> = <font color="Maroon" title="c19">l5</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/><div><span class="lab"><font color="Green" title="E51">A30</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="39_1_5_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">  not <font color="Maroon" title="c19">l5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">r5</font></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:39_1_5_2"/><span class="lab"><font color="Green" title="E52">A31</font></span>: 
<font color="Maroon" title="c19">l5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">r5</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><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="39_1_5_2_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">  not <font color="Maroon" title="c19">l5</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">r5</font></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:39_1_5_2_1"/>
<font color="Maroon" title="c19">l5</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">r5</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="E2:39_1_5_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">r5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">l5</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">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T50" onmouseover="rs('xreal_1/T50')" onmouseout="rh()">XREAL_1:50</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">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="E4:39_1_5_2_1"/><span class="lab"><font color="Green" title="E53">A32</font></span>: 
<font color="Maroon" title="c21">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">P5</font>
 <span class="kw">and </span><br/><a NAME="E5:39_1_5_2_1"/><span class="lab"><font color="Green" title="E54">A33</font></span>: 
<font color="Maroon" title="c21">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">r5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">l5</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39_1_5"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="txt" href="#E10:39_1_5"><span class="lab"><font color="Green" title="E50">A29</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="E6:39_1_5_2_1"/>
 ex <font color="Olive" title="b1">s6</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">s6</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font> &amp; <font color="Olive" title="b1">s6</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c16">r5</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s6</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1_5_2_1"><span class="lab"><font color="Green" title="E53">A32</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:39_1_5_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39_1_5_2_1"><span class="lab"><font color="Green" title="E54">A33</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="E3:39_1_5_2"/><span class="kw">then </span>
<font color="Maroon" title="c19">l5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">r5</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1_5_2"><span class="lab"><font color="Green" title="E52">A31</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">r7</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E5:39_1_5_2"/><span class="lab"><font color="Green" title="E53">A34</font></span>: 
<font color="Maroon" title="c21">r7</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="E6:39_1_5_2"/><span class="lab"><font color="Green" title="E54">A35</font></span>: 
 <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c20">m5</font>,<font color="Maroon" title="c21">r7</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">B0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:39_1"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="txt" href="#E4:39_1_5"><span class="lab"><font color="Green" title="E46">A25</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="c22">r9</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E8:39_1_5_2"/><span class="lab"><font color="Green" title="E55">A36</font></span>: 
<font color="Maroon" title="c22">r9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">P5</font>
 <span class="kw">and </span><br/><a NAME="E9:39_1_5_2"/><span class="lab"><font color="Green" title="E56">A37</font></span>: 
<font color="Maroon" title="c22">r9</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c21">r7</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39_1_5"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="txt" href="#E10:39_1_5"><span class="lab"><font color="Green" title="E50">A29</font></span></a>, <a class="txt" href="#E5:39_1_5_2"><span class="lab"><font color="Green" title="E53">A34</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="c23">r9</font> = <font color="Maroon" title="c22">r9</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="c24">e8</font> = <font color="Maroon" title="c23">r9</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="E12:39_1_5_2"/>
<font color="Maroon" title="c19">l5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">r9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:39_1_5"><span class="lab"><font color="Green" title="E50">A29</font></span></a>, <a class="txt" href="#E8:39_1_5_2"><span class="lab"><font color="Green" title="E55">A36</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="E13:39_1_5_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A38</font></span>: 
<font color="Maroon" title="c23">r9</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c19">l5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a></span>;<br/><a NAME="E14:39_1_5_2"/>
<font color="Maroon" title="c23">r9</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c19">l5</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c21">r7</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">l5</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39_1_5_2"><span class="lab"><font color="Green" title="E56">A37</font></span></a>, <a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><a NAME="E15:39_1_5_2"/><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="c23">r9</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c19">l5</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="c21">r7</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:39_1_5_2"><span class="lab"><font color="Green" title="E57">A38</font></span></a>, <a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>;<br/><a NAME="E16:39_1_5_2"/><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="c24">e8</font>,<font color="Maroon" title="c20">m5</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c21">r7</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="E17:39_1_5_2"/><span class="kw">then </span>
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c24">e8</font>,<font color="Maroon" title="c20">m5</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c21">r7</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="E18:39_1_5_2"/><span class="kw">then </span>
<font color="Maroon" title="c24">e8</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="c20">m5</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="c21">r7</font> </span> } </span> 
 ;<br/><a NAME="E19:39_1_5_2"/><span class="kw">then </span>
<font color="Maroon" title="c24">e8</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="c20">m5</font>,<font color="Maroon" title="c21">r7</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="E20:39_1_5_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_1"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="txt" href="#E7:39_1_5"><span class="lab"><font color="Green" title="E48">A27</font></span></a>, <a class="txt" href="#E6:39_1_5_2"><span class="lab"><font color="Green" title="E54">A35</font></span></a>, <a class="txt" href="#E8:39_1_5_2"><span class="lab"><font color="Green" title="E55">A36</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="E14:39_1_5"/><span class="lab"><font color="Green" title="E52">A39</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">r5</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E8:39_1"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="txt" href="#E4:39_1_5"><span class="lab"><font color="Green" title="E46">A25</font></span></a>, <a class="ref" href="xxreal_1.html#T4" onmouseover="rs('xxreal_1/T4')" onmouseout="rh()">XXREAL_1:4</a></span>;<br/><a NAME="E15:39_1_5"/><span class="lab"><font color="Green" title="E53">A40</font></span>: 
<font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">r5</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">by</span> <span class="lab"><a class="txt" href="#E13:39_1_5"><span class="lab"><font color="Green" title="E51">A30</font></span></a>, <a class="ref" href="xreal_1.html#T50" onmouseover="rs('xreal_1/T50')" onmouseout="rh()">XREAL_1:50</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c21">q</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c17">P5</font>;<br/><a NAME="E16:39_1_5"/><span class="lab"><font color="Green" title="E54">A41</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c17">P5</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c17">P5</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39_1_5"><span class="lab"><font color="Green" title="E49">A28</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c22">q1</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c17">P5</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/><a NAME="E18:39_1_5"/>
<font color="Maroon" title="c22">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:39_1_5"><span class="lab"><font color="Green" title="E48">A27</font></span></a>, <a class="txt" href="#E16:39_1_5"><span class="lab"><font color="Green" title="E54">A41</font></span></a></span>;<br/><a NAME="E19:39_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A42</font></span>: 
<font color="Maroon" title="c22">q1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E8:39_1"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="ref" href="xxreal_1.html#T4" onmouseover="rs('xxreal_1/T4')" onmouseout="rh()">XXREAL_1:4</a></span>;<br/><a NAME="E20:39_1_5"/>
<font color="Maroon" title="c19">l5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39_1_5"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="txt" href="#E10:39_1_5"><span class="lab"><font color="Green" title="E50">A29</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="E21:39_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c19">l5</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:39_1_5"><span class="lab"><font color="Green" title="E55">A42</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="E22:39_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c19">l5</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:39_1_5"><span class="lab"><font color="Green" title="E51">A30</font></span></a>, <a class="txt" href="#E14:39_1_5"><span class="lab"><font color="Green" title="E52">A39</font></span></a>, <a class="ref" href="xxreal_1.html#T4" onmouseover="rs('xxreal_1/T4')" onmouseout="rh()">XXREAL_1:4</a></span>;<br/><a NAME="E23:39_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A43</font></span>: 
( <font color="Maroon" title="c19">l5</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</font> or <font color="Maroon" title="c19">l5</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">Q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E8:39_1"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="39_1_5_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">  not <font color="Maroon" title="c19">l5</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</font></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:39_1_5_3"/>
<font color="Maroon" title="c19">l5</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">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/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">s5</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E3:39_1_5_3"/><span class="lab"><font color="Green" title="E57">A44</font></span>: 
<font color="Maroon" title="c23">s5</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="E4:39_1_5_3"/><span class="lab"><font color="Green" title="E58">A45</font></span>: 
 <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c20">m5</font>,<font color="Maroon" title="c23">s5</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:39_1"><span class="lab"><font color="Green" title="E38">A15</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="c24">s5</font> = <font color="Maroon" title="c23">s5</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">set </span><font color="Maroon" title="c25">s59</font> =  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c24">s5</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">r5</font></span>)</span>);<br/><a NAME="E6:39_1_5_3"/><span class="lab"><font color="Green" title="E59">A46</font></span>: 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c24">s5</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">r5</font></span>)</span>) <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">by</span> <span class="lab"><a class="txt" href="#E15:39_1_5"><span class="lab"><font color="Green" title="E53">A40</font></span></a>, <a class="txt" href="#E3:39_1_5_3"><span class="lab"><font color="Green" title="E57">A44</font></span></a>, <a class="ref" href="xxreal_0.html#T21" onmouseover="rs('xxreal_0/T21')" onmouseout="rh()">XXREAL_0:21</a></span>;<br/><a NAME="E7:39_1_5_3"/><span class="lab"><font color="Green" title="E60">A47</font></span>: 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c24">s5</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">r5</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">s5</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T17" onmouseover="rs('xxreal_0/T17')" onmouseout="rh()">XXREAL_0:17</a></span>;<br/><a NAME="E8:39_1_5_3"/><span class="lab"><font color="Green" title="E61">A48</font></span>: 
<span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c24">s5</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">r5</font></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c24">s5</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">r5</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_1_5_3"><span class="lab"><font color="Green" title="E59">A46</font></span></a>, <a class="ref" href="xreal_1.html#T216" onmouseover="rs('xreal_1/T216')" onmouseout="rh()">XREAL_1:216</a></span>;<br/><a NAME="E9:39_1_5_3"/>
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c24">s5</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">r5</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">r5</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T17" onmouseover="rs('xxreal_0/T17')" onmouseout="rh()">XXREAL_0:17</a></span>;<br/><a NAME="E10:39_1_5_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c24">s5</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">r5</font></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">r5</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:39_1_5_3"><span class="lab"><font color="Green" title="E61">A48</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="E11:39_1_5_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c24">s5</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">r5</font></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c16">r5</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">r5</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c16">r5</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E12:39_1_5_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A49</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c24">s5</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">r5</font></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c16">r5</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c24">s5</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">r5</font></span>)</span>)</span>)</span> <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">l5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c24">s5</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">r5</font></span>)</span>)</span>)</span> <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="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c26">e1</font> = <font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c24">s5</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">r5</font></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</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="c27">e1</font> = <font color="Maroon" title="c26">e1</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="E15:39_1_5_3"/>
<span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c24">s5</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">r5</font></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c24">s5</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">r5</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_1_5_3"><span class="lab"><font color="Green" title="E59">A46</font></span></a>, <a class="ref" href="xreal_1.html#T216" onmouseover="rs('xreal_1/T216')" onmouseout="rh()">XREAL_1:216</a></span>;<br/><a NAME="E16:39_1_5_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c24">s5</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">r5</font></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">s5</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:39_1_5_3"><span class="lab"><font color="Green" title="E60">A47</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="E17:39_1_5_3"/><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="c19">l5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c24">s5</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">r5</font></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></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="c24">s5</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_1_5_3"><span class="lab"><font color="Green" title="E59">A46</font></span></a>, <a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>;<br/><a NAME="E18:39_1_5_3"/><span class="kw">then </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="c19">l5</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c24">s5</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">r5</font></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">s5</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></span>;<br/><a NAME="E19:39_1_5_3"/><span class="kw">then </span>
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c20">m5</font>,<font color="Maroon" title="c27">e1</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">s5</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="E20:39_1_5_3"/><span class="kw">then </span>
<font color="Maroon" title="c27">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="c20">m5</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="c24">s5</font> </span> } </span> 
 ;<br/><a NAME="E21:39_1_5_3"/><span class="kw">then </span>
<font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c24">s5</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">r5</font></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <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="c20">m5</font>,<font color="Maroon" title="c24">s5</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="E22:39_1_5_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A50</font></span>: 
<font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c24">s5</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">r5</font></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">s7</font> where <font color="Olive" title="b1">s7</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : ( <font color="Olive" title="b1">s7</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c16">r5</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s7</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1_5_3"><span class="lab"><font color="Green" title="E58">A45</font></span></a>, <a class="txt" href="#E12:39_1_5_3"><span class="lab"><font color="Green" title="E62">A49</font></span></a></span>;<br/><a NAME="E23:39_1_5_3"/>
<font color="Maroon" title="c19">l5</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c24">s5</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">r5</font></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_1_5_3"><span class="lab"><font color="Green" title="E59">A46</font></span></a>, <a class="ref" href="xreal_1.html#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a>, <a class="ref" href="xreal_1.html#T139" onmouseover="rs('xreal_1/T139')" onmouseout="rh()">XREAL_1:139</a></span>;<br/><a NAME="E24:39_1_5_3"/><span class="kw">then </span>
<font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c24">s5</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">r5</font></span>)</span>)</span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c24">s5</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">r5</font></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c24">s5</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">r5</font></span>)</span>)</span>)</span> <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="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><span class="kw">hence </span><a NAME="E25:39_1_5_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:39_1_5"><span class="lab"><font color="Green" title="E50">A29</font></span></a>, <a class="txt" href="#E22:39_1_5_3"><span class="lab"><font color="Green" title="E63">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><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">s1</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E26:39_1_5"/><span class="lab"><font color="Green" title="E57">A51</font></span>: 
<font color="Maroon" title="c23">s1</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="E27:39_1_5"/><span class="lab"><font color="Green" title="E58">A52</font></span>: 
 <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c20">m5</font>,<font color="Maroon" title="c23">s1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">B0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:39_1"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="txt" href="#E23:39_1_5"><span class="lab"><font color="Green" title="E56">A43</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/><a NAME="E28:39_1_5"/>
<font color="Maroon" title="c23">s1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <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">by</span> <span class="lab"><a class="txt" href="#E26:39_1_5"><span class="lab"><font color="Green" title="E57">A51</font></span></a>, <a class="ref" href="xreal_1.html#T139" onmouseover="rs('xreal_1/T139')" onmouseout="rh()">XREAL_1:139</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">r2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E30:39_1_5"/><span class="lab"><font color="Green" title="E59">A53</font></span>: 
<font color="Maroon" title="c24">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">P5</font>
 <span class="kw">and </span><br/><a NAME="E31:39_1_5"/><span class="lab"><font color="Green" title="E60">A54</font></span>: 
<font color="Maroon" title="c24">r2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">s1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39_1_5"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="txt" href="#E10:39_1_5"><span class="lab"><font color="Green" title="E50">A29</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="c25">r2</font> = <font color="Maroon" title="c24">r2</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="E33:39_1_5"/><span class="lab"><font color="Green" title="E61">A55</font></span>: 
<font color="Maroon" title="c19">l5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:39_1_5"><span class="lab"><font color="Green" title="E50">A29</font></span></a>, <a class="txt" href="#E30:39_1_5"><span class="lab"><font color="Green" title="E59">A53</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="c26">s3</font> = <font color="Maroon" title="c25">r2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c19">l5</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>  ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c27">e1</font> = <font color="Maroon" title="c19">l5</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c26">s3</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="E36:39_1_5"/><span class="lab"><font color="Green" title="E62">A56</font></span>: 
<font color="Maroon" title="c25">r2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c19">l5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:39_1_5"><span class="lab"><font color="Green" title="E61">A55</font></span></a>, <a class="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a></span>;<br/><a NAME="E37:39_1_5"/><span class="lab"><font color="Green" title="E63">A57</font></span>: 
<font color="Maroon" title="c25">r2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c19">l5</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">l5</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">s1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">l5</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:39_1_5"><span class="lab"><font color="Green" title="E60">A54</font></span></a>, <a class="ref" href="xreal_1.html#T14" onmouseover="rs('xreal_1/T14')" onmouseout="rh()">XREAL_1:14</a></span>;<br/><a NAME="E38:39_1_5"/>
<font color="Maroon" title="c23">s1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">s1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:39_1_5"><span class="lab"><font color="Green" title="E57">A51</font></span></a>, <a class="ref" href="xreal_1.html#T216" onmouseover="rs('xreal_1/T216')" onmouseout="rh()">XREAL_1:216</a></span>;<br/><a NAME="E39:39_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A58</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">l5</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c26">s3</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c19">l5</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">s1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:39_1_5"><span class="lab"><font color="Green" title="E63">A57</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="E40:39_1_5"/>
<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"><font color="Maroon" title="c19">l5</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c26">s3</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c19">l5</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">l5</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c26">s3</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c19">l5</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:39_1_5"><span class="lab"><font color="Green" title="E62">A56</font></span></a>, <a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>;<br/><a NAME="E41:39_1_5"/><span class="kw">then </span>
<a href="metric_1.html#K6" title="METRIC_1:func.6">real_dist</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c19">l5</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c26">s3</font></span>)</span>,<font color="Maroon" title="c19">l5</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">s1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:39_1_5"><span class="lab"><font color="Green" title="E64">A58</font></span></a>, <a class="ref" href="metric_1.html#D12" onmouseover="rs('metric_1/D12')" onmouseout="rh()">METRIC_1:def 12</a></span>;<br/><a NAME="E42:39_1_5"/><span class="kw">then </span>
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c27">e1</font>,<font color="Maroon" title="c20">m5</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">s1</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="E43:39_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c19">l5</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c26">s3</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="c20">m5</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="c23">s1</font> </span> } </span> 
 ;<br/><a NAME="E44:39_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c19">l5</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c26">s3</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="c20">m5</font>,<font color="Maroon" title="c23">s1</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="E45:39_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A59</font></span>: 
<font color="Maroon" title="c19">l5</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c26">s3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c17">P5</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:39_1_5"><span class="lab"><font color="Green" title="E58">A52</font></span></a>, <a class="txt" href="#E30:39_1_5"><span class="lab"><font color="Green" title="E59">A53</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E46:39_1_5"/>
<font color="Maroon" title="c17">P5</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:39_1_5"><span class="lab"><font color="Green" title="E48">A27</font></span></a>, <a class="ref" href="xboole_1.html#T26" onmouseover="rs('xboole_1/T26')" onmouseout="rh()">XBOOLE_1:26</a></span>;<br/><span class="kw">hence </span><a NAME="E47:39_1_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_1"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="txt" href="#E45:39_1_5"><span class="lab"><font color="Green" title="E65">A59</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="c15">q</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">Q</font>;<br/>
<a NAME="E31:39_1"/><span class="lab"><font color="Green" title="E45">A60</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">Q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">Q</font>
 
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">q1</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">Q</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<a NAME="E33:39_1"/><span class="lab"><font color="Green" title="E46">A61</font></span>: 
<font color="Maroon" title="c16">q1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E8:39_1"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="txt" href="#E31:39_1"><span class="lab"><font color="Green" title="E45">A60</font></span></a>, <a class="ref" href="xxreal_1.html#T4" onmouseover="rs('xxreal_1/T4')" onmouseout="rh()">XXREAL_1:4</a></span>;<br/>
<a NAME="E34:39_1"/>
<font color="Maroon" title="c12">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">q1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:39_1"><span class="lab"><font color="Green" title="E42">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/>
<a NAME="E35:39_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:39_1"><span class="lab"><font color="Green" title="E46">A61</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="E36:39_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:39_1"><span class="lab"><font color="Green" title="E44">A23</font></span></a>, <a class="ref" href="xxreal_1.html#T4" onmouseover="rs('xxreal_1/T4')" onmouseout="rh()">XXREAL_1:4</a></span>;<br/>
<a NAME="E37:39_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A62</font></span>: 
<font color="Maroon" title="c14">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">A0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E8:39_1"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="txt" href="#E29:39_1"><span class="lab"><font color="Green" title="E43">A20</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E38:39_1"/>
<font color="Maroon" title="c8">A0</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="txt" href="#E15:39_1"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="ref" href="tsep_1.html#T17" onmouseover="rs('tsep_1/T17')" onmouseout="rh()">TSEP_1:17</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">s1</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E40:39_1"/><span class="lab"><font color="Green" title="E48">A63</font></span>: 
<font color="Maroon" title="c17">s1</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="E41:39_1"/><span class="lab"><font color="Green" title="E49">A64</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="c17">s1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">A0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:39_1"><span class="lab"><font color="Green" title="E47">A62</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/>
<a NAME="E42:39_1"/>
<font color="Maroon" title="c17">s1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <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">by</span> <span class="lab"><a class="txt" href="#E40:39_1"><span class="lab"><font color="Green" title="E48">A63</font></span></a>, <a class="ref" href="xreal_1.html#T139" onmouseover="rs('xreal_1/T139')" onmouseout="rh()">XREAL_1:139</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">r2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E44:39_1"/><span class="lab"><font color="Green" title="E50">A65</font></span>: 
<font color="Maroon" title="c18">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Q</font>
 <span class="kw">and </span><br/><a NAME="E45:39_1"/><span class="lab"><font color="Green" title="E51">A66</font></span>: 
<font color="Maroon" title="c18">r2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">l</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">s1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:39_1"><span class="lab"><font color="Green" title="E42">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="c19">r2</font> = <font color="Maroon" title="c18">r2</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="E47:39_1"/><span class="lab"><font color="Green" title="E52">A67</font></span>: 
<font color="Maroon" title="c12">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">r2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:39_1"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="txt" href="#E44:39_1"><span class="lab"><font color="Green" title="E50">A65</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">set </span><font color="Maroon" title="c20">s3</font> = <font color="Maroon" title="c19">r2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c12">l</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">e1</font> = <font color="Maroon" title="c12">l</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">r2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c12">l</font></span>)</span> 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="E49:39_1"/><span class="lab"><font color="Green" title="E53">A68</font></span>: 
<font color="Maroon" title="c19">r2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c12">l</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:39_1"><span class="lab"><font color="Green" title="E52">A67</font></span></a>, <a class="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a></span>;<br/>
<a NAME="E50:39_1"/><span class="lab"><font color="Green" title="E54">A69</font></span>: 
<font color="Maroon" title="c19">r2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c12">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">s1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:39_1"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="ref" href="xreal_1.html#T14" onmouseover="rs('xreal_1/T14')" onmouseout="rh()">XREAL_1:14</a></span>;<br/>
<a NAME="E51:39_1"/>
<font color="Maroon" title="c17">s1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">s1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:39_1"><span class="lab"><font color="Green" title="E48">A63</font></span></a>, <a class="ref" href="xreal_1.html#T216" onmouseover="rs('xreal_1/T216')" onmouseout="rh()">XREAL_1:216</a></span>;<br/>
<a NAME="E52:39_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A70</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">r2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c12">l</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c12">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">s1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:39_1"><span class="lab"><font color="Green" title="E54">A69</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="E53:39_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"><font color="Maroon" title="c12">l</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">r2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c12">l</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c12">l</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">r2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c12">l</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c12">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:39_1"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>;<br/>
<a NAME="E54:39_1"/><span class="kw">then </span>
<a href="metric_1.html#K6" title="METRIC_1:func.6">real_dist</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">r2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c12">l</font></span>)</span></span>)</span>,<font color="Maroon" title="c12">l</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">s1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:39_1"><span class="lab"><font color="Green" title="E55">A70</font></span></a>, <a class="ref" href="metric_1.html#D12" onmouseover="rs('metric_1/D12')" onmouseout="rh()">METRIC_1:def 12</a></span>;<br/>
<a NAME="E55:39_1"/><span class="kw">then </span>
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c21">e1</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="c17">s1</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="E56:39_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">l</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">r2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c12">l</font></span>)</span> <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="c17">s1</font> </span> } </span> 
 
;<br/>
<a NAME="E57:39_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">l</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">r2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c12">l</font></span>)</span> <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="c17">s1</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="E58:39_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_1"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="txt" href="#E41:39_1"><span class="lab"><font color="Green" title="E49">A64</font></span></a>, <a class="txt" href="#E44:39_1"><span class="lab"><font color="Green" title="E50">A65</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="E6:39"/><span class="kw">then </span>
<a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P7</font> 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#T11" onmouseover="rs('connsp_1/T11')" onmouseout="rh()">CONNSP_1:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:39"/>
( <font color="Maroon" title="c1">P7</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c1">P7</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E2:39"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E26">A3</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="connsp_1.html#D3" onmouseover="rs('connsp_1/D3')" onmouseout="rh()">CONNSP_1:def 3</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>
