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

<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp;  <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b2">s</font></span>)</span> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c3">At</font> = <font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c4">Bt</font> = <font color="Maroon" title="c2">B</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</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>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a>, <a class="ref" href="topreal6.html#T81" title="TOPREAL6:th.81">TOPREAL6:81</a></i>;<br/>
<a NAME="E2:13"/><i><font color="Green" title="E11">A1</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><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="c3">At</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">At</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><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">Bt</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Bt</font> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">AB</font> = <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><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="c3">At</font></span>)</span>,<span class="p2">(<span class="default"><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">Bt</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b2">s</font></span>)</span> );<br/>
<div><i><font color="Green" title="E12">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">AB</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">t</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Olive" title="b1">t</font>]</span><br/><a NAME="E1:13_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">AB</font>
 ;<br/><a NAME="E2:13_1"/><b>then </b>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 <b>by </b><i><a class="txt" href="#E2:13"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">r</font>, <font color="Maroon" title="c8">s</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:13_1"/><i><font color="Green" title="E13">A3</font></i>: 
( <font color="Maroon" title="c7">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 <b>and </b><br/><a NAME="E5:13_1"/><i><font color="Green" title="E14">A4</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c8">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T103" title="ZFMISC_1:th.103">ZFMISC_1:103</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">r</font> = <font color="Maroon" title="c7">r</font>, <font color="Maroon" title="c10">s</font> = <font color="Maroon" title="c8">s</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="#E4:13_1"><i><font color="Green" title="E13">A3</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c11">t</font> =  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">s</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c11">t</font>]</span><br/><b>thus </b><a NAME="E7:13_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c11">t</font>]
 <b>by </b><i><a class="txt" href="#E5:13_1"><i><font color="Green" title="E14">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c6">f</font> being   <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <font color="Maroon" title="c5">AB</font><b> such that </b><br/><a NAME="E6:13"/><i><font color="Green" title="E13">A5</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">AB</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c6">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E4:13"><i><font color="Green" title="E12">A2</font></i></a>);<br/></i>
<a NAME="E7:13"/>
for <font color="Olive" title="b1">Y</font> being  <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>   st <font color="Olive" title="b1">Y</font> is <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> holds <br/><font color="Maroon" title="c6">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">Y</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">Y</font> be   <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">Y</font> is <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> implies <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">Y</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )</span><br/>

<b>assume </b><a NAME="E1:13_2"/><i><font color="Green" title="E14">A6</font></i>: 
<font color="Maroon" title="c7">Y</font> is <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">Y</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a></span><br/>

<a NAME="E2:13_2"/>
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c5">AB</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">Y</font> holds <br/> ex <font color="Olive" title="b2">YS</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><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="c3">At</font></span>)</span> ex <font color="Olive" title="b3">YT</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><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">Bt</font></span>)</span> st <br/>( <font color="Olive" title="b2">YS</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b3">YT</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b2">YS</font>,<font color="Olive" title="b3">YT</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> &amp; <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b2">YS</font>,<font color="Olive" title="b3">YT</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">Y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c5">AB</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">Y</font> implies  ex <font color="Olive" title="b1">YS</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><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="c3">At</font></span>)</span> ex <font color="Olive" title="b2">YT</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><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">Bt</font></span>)</span> st <br/>( <font color="Olive" title="b1">YS</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">YT</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">YS</font>,<font color="Olive" title="b2">YT</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> &amp; <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">YS</font>,<font color="Olive" title="b2">YT</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">Y</font> ) )</span><br/>

<b>assume </b><a NAME="E1:13_2_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">YS</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><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="c3">At</font></span>)</span> ex <font color="Olive" title="b2">YT</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><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">Bt</font></span>)</span> st <br/>( <font color="Olive" title="b1">YS</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">YT</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">YS</font>,<font color="Olive" title="b2">YT</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> &amp; <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">YS</font>,<font color="Olive" title="b2">YT</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">Y</font> )</span><br/>

<a NAME="E2:13_2_1"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">N</font> being    <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c6">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">x</font><b> such that </b><br/><a NAME="E4:13_2_1"/><i><font color="Green" title="E15">A7</font></i>: 
<font color="Maroon" title="c9">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Y</font>
 <b>by </b><i><a class="txt" href="#E1:13_2"><i><font color="Green" title="E14">A6</font></i></a>, <a class="ref" href="rcomp_1.html#T39" title="RCOMP_1:th.39">RCOMP_1:39</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">g</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E6:13_2_1"/><i><font color="Green" title="E16">A8</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">g</font>
 <b>and </b><br/><a NAME="E7:13_2_1"/><i><font color="Green" title="E17">A9</font></i>: 
<font color="Maroon" title="c9">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c10">g</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D7" title="RCOMP_1:def.7">RCOMP_1:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">r</font>, <font color="Maroon" title="c12">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E9:13_2_1"/><i><font color="Green" title="E18">A10</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">r</font>,<font color="Maroon" title="c12">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E10:13_2_1"/><i><font color="Green" title="E19">A11</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">s</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E6:13"><i><font color="Green" title="E13">A5</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">a</font> = <font color="Maroon" title="c11">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>, <font color="Maroon" title="c14">b</font> = <font color="Maroon" title="c11">r</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>, <font color="Maroon" title="c15">c</font> = <font color="Maroon" title="c12">s</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>, <font color="Maroon" title="c16">d</font> = <font color="Maroon" title="c12">s</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c17">S</font> = <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c14">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>, <font color="Maroon" title="c18">T</font> = <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c15">c</font>,<font color="Maroon" title="c16">d</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</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>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">YS</font> = <font color="Maroon" title="c17">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c1">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><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="c3">At</font></span>)</span> <b>by </b><i><a class="txt" href="#E2:13"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">YT</font> = <font color="Maroon" title="c18">T</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">B</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><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">Bt</font></span>)</span> <b>by </b><i><a class="txt" href="#E2:13"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c19">YS</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">YT</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><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">Bt</font></span>)</span> st <br/>( <font color="Maroon" title="c19">YS</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b1">YT</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c19">YS</font>,<font color="Olive" title="b1">YT</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> &amp; <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c19">YS</font>,<font color="Olive" title="b1">YT</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">Y</font> )</span><br/>

<b>take </b>
<font color="Maroon" title="c20">YT</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c19">YS</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c20">YT</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c19">YS</font>,<font color="Maroon" title="c20">YT</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> &amp; <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c19">YS</font>,<font color="Maroon" title="c20">YT</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">Y</font> )</span><br/>

<a NAME="E15:13_2_1"/>
( <font color="Maroon" title="c17">S</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c18">T</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 
<b>by </b><i><a class="ref" href="jordan6.html#T46" title="JORDAN6:th.46">JORDAN6:46</a></i>;<br/>
<b>hence </b><a NAME="E16:13_2_1"/>
( <font color="Maroon" title="c19">YS</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c20">YT</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <b>by </b><i><a class="txt" href="#E2:13"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="tsp_1.html#D1" title="TSP_1:def.1">TSP_1:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c19">YS</font>,<font color="Maroon" title="c20">YT</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> &amp; <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c19">YS</font>,<font color="Maroon" title="c20">YT</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">Y</font> )</span><br/>

<a NAME="E17:13_2_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">g</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 
<b>by </b><i><a class="txt" href="#E6:13_2_1"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="xreal_1.html#T217" title="XREAL_1:th.217">XREAL_1:217</a></i>;<br/>
<a NAME="E18:13_2_1"/><b>then </b><i><font color="Green" title="E20">A12</font></i>: 
( <font color="Maroon" title="c11">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">S</font> &amp; <font color="Maroon" title="c12">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">T</font> )
 
<b>by </b><i><a class="ref" href="topreal6.html#T20" title="TOPREAL6:th.20">TOPREAL6:20</a></i>;<br/>
<a NAME="E19:13_2_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">AB</font>
 
;<br/>
<a NAME="E20:13_2_1"/><b>then </b>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E2:13"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E21:13_2_1"/><b>then </b>
( <font color="Maroon" title="c11">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> )
 
<b>by </b><i><a class="txt" href="#E9:13_2_1"><i><font color="Green" title="E18">A10</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E22:13_2_1"/><b>then </b>
( <font color="Maroon" title="c11">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">YS</font> &amp; <font color="Maroon" title="c12">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">YT</font> )
 
<b>by </b><i><a class="txt" href="#E18:13_2_1"><i><font color="Green" title="E20">A12</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E23:13_2_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c19">YS</font>,<font color="Maroon" title="c20">YT</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E9:13_2_1"><i><font color="Green" title="E18">A10</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c19">YS</font>,<font color="Maroon" title="c20">YT</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">Y</font></span><br/>

<a NAME="E24:13_2_1"/><i><font color="Green" title="E21">A13</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</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 <font color="Maroon" title="c5">AB</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E25:13_2_1"/>
<font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c19">YS</font>,<font color="Maroon" title="c20">YT</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">N</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c21">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c19">YS</font>,<font color="Maroon" title="c20">YT</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> or <font color="Maroon" title="c21">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">N</font> )</span><br/>

<b>assume </b><a NAME="E1:13_2_1_1"/>
<font color="Maroon" title="c21">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c19">YS</font>,<font color="Maroon" title="c20">YT</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c21">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">N</font></span><br/>

<b>then consider </b><font color="Maroon" title="c22">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">AB</font><b> such that </b><br/><a NAME="E3:13_2_1_1"/><i><font color="Green" title="E22">A14</font></i>: 
<font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c19">YS</font>,<font color="Maroon" title="c20">YT</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>and </b><br/><a NAME="E4:13_2_1_1"/><i><font color="Green" title="E23">A15</font></i>: 
<font color="Maroon" title="c21">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">y</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c23">r1</font>, <font color="Maroon" title="c24">s1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E6:13_2_1_1"/><i><font color="Green" title="E24">A16</font></i>: 
<font color="Maroon" title="c22">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">r1</font>,<font color="Maroon" title="c24">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E7:13_2_1_1"/><i><font color="Green" title="E25">A17</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">r1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c24">s1</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E6:13"><i><font color="Green" title="E13">A5</font></i></a></i>;<br/>
<a NAME="E8:13_2_1_1"/>
<font color="Maroon" title="c23">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">YS</font>
 
<b>by </b><i><a class="txt" href="#E3:13_2_1_1"><i><font color="Green" title="E22">A14</font></i></a>, <a class="txt" href="#E6:13_2_1_1"><i><font color="Green" title="E24">A16</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E9:13_2_1_1"/><b>then </b>
<font color="Maroon" title="c23">r1</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"><span class="p2">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E10:13_2_1_1"/><b>then </b><i><font color="Green" title="E26">A18</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">r1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">r</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">g</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T8" title="RCOMP_1:th.8">RCOMP_1:8</a></i>;<br/>
<a NAME="E11:13_2_1_1"/>
<font color="Maroon" title="c24">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">YT</font>
 
<b>by </b><i><a class="txt" href="#E3:13_2_1_1"><i><font color="Green" title="E22">A14</font></i></a>, <a class="txt" href="#E6:13_2_1_1"><i><font color="Green" title="E24">A16</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E12:13_2_1_1"/><b>then </b>
<font color="Maroon" title="c24">s1</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"><span class="p2">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E13:13_2_1_1"/><b>then </b><i><font color="Green" title="E27">A19</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">s1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">g</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T8" title="RCOMP_1:th.8">RCOMP_1:8</a></i>;<br/>
<a NAME="E14:13_2_1_1"/>
<font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>
 
;<br/>
<a NAME="E15:13_2_1_1"/><b>then </b><i><font color="Green" title="E28">A20</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">r1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">r</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c24">s1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">s</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">g</font>
 
<b>by </b><i><a class="txt" href="#E10:13_2_1_1"><i><font color="Green" title="E26">A18</font></i></a>, <a class="txt" href="#E13:13_2_1_1"><i><font color="Green" title="E27">A19</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E16:13_2_1_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">r1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c24">s1</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">s</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">r1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">r</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c24">s1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">s</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jct_misc.html#T11" target="_self" title="JCT_MISC:th.11">Th11</a></i>;<br/>
<a NAME="E17:13_2_1_1"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">r1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c24">s1</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">s</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">g</font>
 
<b>by </b><i><a class="txt" href="#E15:13_2_1_1"><i><font color="Green" title="E28">A20</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E18:13_2_1_1"/>
<font color="Maroon" title="c21">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">N</font>
 <b>by </b><i><a class="txt" href="#E7:13_2_1"><i><font color="Green" title="E17">A9</font></i></a>, <a class="txt" href="#E10:13_2_1"><i><font color="Green" title="E19">A11</font></i></a>, <a class="txt" href="#E4:13_2_1_1"><i><font color="Green" title="E23">A15</font></i></a>, <a class="txt" href="#E7:13_2_1_1"><i><font color="Green" title="E25">A17</font></i></a>, <a class="ref" href="rcomp_1.html#T8" title="RCOMP_1:th.8">RCOMP_1:8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E26:13_2_1"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c19">YS</font>,<font color="Maroon" title="c20">YT</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Y</font>
 
<b>by </b><i><a class="txt" href="#E4:13_2_1"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E27:13_2_1"/>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c19">YS</font>,<font color="Maroon" title="c20">YT</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">Y</font>
 <b>by </b><i><a class="txt" href="#E24:13_2_1"><i><font color="Green" title="E21">A13</font></i></a>, <a class="ref" href="funct_1.html#T163" title="FUNCT_1:th.163">FUNCT_1:163</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:13_2"/>
<font color="Maroon" title="c6">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">Y</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="jct_misc.html#T13" target="_self" title="JCT_MISC:th.13">Th13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">f</font> = <font color="Maroon" title="c6">f</font> as  <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a> <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <font color="Maroon" title="c5">AB</font> <b>by </b><i><a class="ref" href="pscomp_1.html#T54" title="PSCOMP_1:th.54">PSCOMP_1:54</a></i>;<br/>
<a NAME="E9:13"/>
<font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">AB</font> is <a href="pscomp_1.html#V2" title="PSCOMP_1:attr.2">with_min</a>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D15" title="PSCOMP_1:def.15">PSCOMP_1:def 15</a></i>;<br/>
<a NAME="E10:13"/><b>then </b>
 <a href="pscomp_1.html#NK3" title="PSCOMP_1:NK.3">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">AB</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">AB</font>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D4" title="PSCOMP_1:def.4">PSCOMP_1:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">AB</font><b> such that </b><br/><a NAME="E12:13"/><i><font color="Green" title="E14">A21</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">AB</font>
 <b>and </b><br/><a NAME="E13:13"/><i><font color="Green" title="E15">A22</font></i>: 
 <a href="pscomp_1.html#NK3" title="PSCOMP_1:NK.3">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">AB</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<a NAME="E14:13"/><i><font color="Green" title="E16">A23</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E2:13"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="#E12:13"><i><font color="Green" title="E14">A21</font></i></a>, <a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">r1</font>, <font color="Maroon" title="c10">s1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E16:13"/><i><font color="Green" title="E17">A24</font></i>: 
( <font color="Maroon" title="c9">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c10">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 <b>and </b><br/><a NAME="E17:13"/><i><font color="Green" title="E18">A25</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">r1</font>,<font color="Maroon" title="c10">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T103" title="ZFMISC_1:th.103">ZFMISC_1:103</a></i>;<br/>
<a NAME="E18:13"/>
( <font color="Maroon" title="c9">r1</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> &amp; <font color="Maroon" title="c10">s1</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> )
 
<b>by </b><i><a class="txt" href="#E16:13"><i><font color="Green" title="E17">A24</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">r1</font> = <font color="Maroon" title="c9">r1</font>, <font color="Maroon" title="c12">s1</font> = <font color="Maroon" title="c10">s1</font> as  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ;<br/>
<b>take </b>
<font color="Maroon" title="c11">r1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">s</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Maroon" title="c11">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp;  <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">s</font></span>)</span> )</span><br/>

<b>take </b>
<font color="Maroon" title="c12">s1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp;  <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">s1</font></span>)</span> )</span><br/>

<b>thus </b><a NAME="E20:13"/>
( <font color="Maroon" title="c11">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> )
 <b>by </b><i><a class="txt" href="#E14:13"><i><font color="Green" title="E16">A23</font></i></a>, <a class="txt" href="#E17:13"><i><font color="Green" title="E18">A25</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">s1</font></span>)</span></span><br/>

<a NAME="E21:13"/><i><font color="Green" title="E19">A26</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">AB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b2">s</font></span>)</span></span>)</span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is    <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>  : ( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b2">s</font></span>)</span></span>)</span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is    <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>  : ( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">AB</font></span>
<div class="add"><b>let </b><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">AB</font> implies <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b2">s</font></span>)</span></span>)</span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is    <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>  : ( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> ) </span>}</span>  )</span><br/><b>assume </b><a NAME="E1:13_3_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">AB</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b2">s</font></span>)</span></span>)</span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is    <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>  : ( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> ) </span>}</span> </span><br/><b>then consider </b><font color="Maroon" title="c14">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">AB</font><b> such that </b><br/><a NAME="E3:13_3_1"/><i><font color="Green" title="E20">A27</font></i>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">AB</font>
 <b>and </b><br/><a NAME="E4:13_3_1"/><i><font color="Green" title="E21">A28</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">y</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/><b>consider </b><font color="Maroon" title="c15">r1</font>, <font color="Maroon" title="c16">s1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E6:13_3_1"/><i><font color="Green" title="E22">A29</font></i>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">r1</font>,<font color="Maroon" title="c16">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E7:13_3_1"/><i><font color="Green" title="E23">A30</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">r1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">s1</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E6:13"><i><font color="Green" title="E13">A5</font></i></a></i>;<br/><a NAME="E8:13_3_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">r1</font>,<font color="Maroon" title="c16">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 <b>by </b><i><a class="txt" href="#E2:13"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="#E3:13_3_1"><i><font color="Green" title="E20">A27</font></i></a>, <a class="txt" href="#E6:13_3_1"><i><font color="Green" title="E22">A29</font></i></a>, <a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/><a NAME="E9:13_3_1"/><b>then </b>
( <font color="Maroon" title="c15">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c16">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E10:13_3_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b2">s</font></span>)</span></span>)</span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is    <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>  : ( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="#E4:13_3_1"><i><font color="Green" title="E21">A28</font></i></a>, <a class="txt" href="#E7:13_3_1"><i><font color="Green" title="E23">A30</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b2">s</font></span>)</span></span>)</span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is    <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>  : ( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> ) </span>}</span>  or <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">AB</font> )</span><br/>

<b>assume </b><a NAME="E2:13_3"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b2">s</font></span>)</span></span>)</span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is    <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>  : ( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">AB</font></span><br/>

<b>then consider </b><font color="Maroon" title="c14">r</font>, <font color="Maroon" title="c15">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:13_3"/><i><font color="Green" title="E20">A31</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c15">s</font></span>)</span>
 <b>and </b><br/><a NAME="E5:13_3"/><i><font color="Green" title="E21">A32</font></i>: 
( <font color="Maroon" title="c14">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> )
 ;<br/>
<a NAME="E6:13_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">r</font>,<font color="Maroon" title="c15">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E5:13_3"><i><font color="Green" title="E21">A32</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">y</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">r</font>,<font color="Maroon" title="c15">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">AB</font> <b>by </b><i><a class="txt" href="#E2:13"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">r1</font>, <font color="Maroon" title="c18">s1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E9:13_3"/><i><font color="Green" title="E22">A33</font></i>: 
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">r1</font>,<font color="Maroon" title="c18">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E10:13_3"/><i><font color="Green" title="E23">A34</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">r1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">s1</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E6:13"><i><font color="Green" title="E13">A5</font></i></a></i>;<br/>
<a NAME="E11:13_3"/>
( <font color="Maroon" title="c17">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">r</font> &amp; <font color="Maroon" title="c18">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">s</font> )
 
<b>by </b><i><a class="txt" href="#E9:13_3"><i><font color="Green" title="E22">A33</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E12:13_3"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">AB</font>
 <b>by </b><i><a class="txt" href="#E4:13_3"><i><font color="Green" title="E20">A31</font></i></a>, <a class="txt" href="#E10:13_3"><i><font color="Green" title="E23">A34</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c13">r</font>, <font color="Maroon" title="c14">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E23:13"/><i><font color="Green" title="E20">A35</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">r</font>,<font color="Maroon" title="c14">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E24:13"/><i><font color="Green" title="E21">A36</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">s</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E6:13"><i><font color="Green" title="E13">A5</font></i></a></i>;<br/>
<a NAME="E25:13"/>
( <font color="Maroon" title="c11">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">r</font> &amp; <font color="Maroon" title="c12">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s</font> )
 
<b>by </b><i><a class="txt" href="#E17:13"><i><font color="Green" title="E18">A25</font></i></a>, <a class="txt" href="#E23:13"><i><font color="Green" title="E20">A35</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E26:13"/>
 <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">s1</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E13:13"><i><font color="Green" title="E15">A22</font></i></a>, <a class="txt" href="#E21:13"><i><font color="Green" title="E19">A26</font></i></a>, <a class="txt" href="#E24:13"><i><font color="Green" title="E21">A36</font></i></a>, <a class="ref" href="jct_misc.html#D2" target="_self" title="JCT_MISC:def.2">Def2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
