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

<span class="kw">set </span><font color="Maroon" title="c2">R</font> =  <a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a> ;<br/>
<span class="kw">defpred </span><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> ] <span class="kw">means </span> ex <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  );<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">x2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> );<br/>
<span class="kw">let </span><font color="Maroon" title="c3">r</font> be   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">o</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><br/> for <font color="Olive" title="b2">f</font> being  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Olive" title="b1">o</font>,<font color="Maroon" title="c3">r</font></span>)</span>,<span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Olive" title="b1">o</font>,<font color="Maroon" title="c3">r</font></span>)</span>  st <font color="Olive" title="b2">f</font> <a href="abian.html#NR4" title="ABIAN:NR.4">has_no_fixpoint</a>  holds <br/> <a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Olive" title="b2">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">o</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span>,<span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span>  st <font color="Olive" title="b1">f</font> <a href="abian.html#NR4" title="ABIAN:NR.4">has_no_fixpoint</a>  holds <br/> <a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Olive" title="b1">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> </span><br/>



<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">x2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> );<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">x2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> );<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">rr</font> = <font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a>  as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">f1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c5">rr</font>;<br/>
<a NAME="E2:22"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>  ex <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">r</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">r</font>]</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c8">x1</font>, <font color="Maroon" title="c9">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E2:22_1"/><span class="lab"><font color="Green" title="E16">A2</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c9">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T50" title="BORSUK_1:th.50">BORSUK_1:50</a></span>;<br/>
<span class="kw">take </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">x2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c7">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">x2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span>]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c8">x1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c8">x1</font>,<font color="Olive" title="b1">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">x2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c9">x2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c9">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">x2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E3:22_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c9">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">x2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c7">xo</font> being   <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E4:22"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c7">xo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E15">A1</font></span></a>);<br/></span>
<a NAME="E5:22"/><span class="lab"><font color="Green" title="E17">A4</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>  ex <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">r</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Olive" title="b1">r</font>]</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c9">x1</font>, <font color="Maroon" title="c10">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E2:22_2"/><span class="lab"><font color="Green" title="E18">A5</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T50" title="BORSUK_1:th.50">BORSUK_1:50</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c10">x2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c10">x2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> ]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c9">x1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c9">x1</font>,<font color="Olive" title="b1">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> &amp; <font color="Maroon" title="c10">x2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c10">x2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> &amp; <font color="Maroon" title="c10">x2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E3:22_2"/>
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> &amp; <font color="Maroon" title="c10">x2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_2"><span class="lab"><font color="Green" title="E18">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c8">fx2</font> being   <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E7:22"/><span class="lab"><font color="Green" title="E18">A6</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c8">fx2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E5:22"><span class="lab"><font color="Green" title="E17">A4</font></span></a>);<br/></span>
<a NAME="E8:22"/><span class="lab"><font color="Green" title="E19">A7</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>  ex <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">r</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">r</font>]</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c10">x1</font>, <font color="Maroon" title="c11">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E2:22_3"/><span class="lab"><font color="Green" title="E20">A8</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c10">x1</font>,<font color="Maroon" title="c11">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T50" title="BORSUK_1:th.50">BORSUK_1:50</a></span>;<br/>
<span class="kw">take </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">x2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c9">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">x2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span>]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c10">x1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c10">x1</font>,<font color="Olive" title="b1">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">x2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c11">x2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c10">x1</font>,<font color="Maroon" title="c11">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">x2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">x2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E3:22_3"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c10">x1</font>,<font color="Maroon" title="c11">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">x2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">x2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_3"><span class="lab"><font color="Green" title="E20">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c9">yo</font> being   <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E10:22"/><span class="lab"><font color="Green" title="E20">A9</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c9">yo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E8:22"><span class="lab"><font color="Green" title="E19">A7</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c10">f1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c5">rr</font> as   <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a>  <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="brouwer.html#E18"><span class="lab"><font color="Green" title="E13">Lm1</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">D2</font> =  <a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">S1</font> =  <a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">OK</font> = <span class="p1">(<span class="default"><a href="brouwer.html#K1" title="BROUWER:func.1">DiffElems</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span>,<span class="p2">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">s</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span>;<br/>
<span class="lab"><font color="Green" title="E21">A10</font></span>: <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">o</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> = 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T46" title="EUCLID:th.46">EUCLID:46</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="toprns_1.html#T24" title="TOPRNS_1:th.24">TOPRNS_1:24</a></span>
;<br/>
<a NAME="E14:22"/><span class="lab"><font color="Green" title="E22">A11</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="toprealb.html#T9" title="TOPREALB:th.9">TOPREALB:9</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E23">A12</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_5"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:22_5"/><span class="lab"><font color="Green" title="E24">A13</font></span>: 
<font color="Maroon" title="c4">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:22_5"/>
(  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22"><span class="lab"><font color="Green" title="E21">A10</font></span></a>, <a class="ref" href="topreal9.html#T7" title="TOPREAL9:th.7">TOPREAL9:7</a>, <a class="ref" href="topreal9.html#T19" title="TOPREAL9:th.19">TOPREAL9:19</a></span>;<br/><span class="kw">hence </span><a NAME="E3:22_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:22"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="txt" href="#E1:22_5"><span class="lab"><font color="Green" title="E24">A13</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E16:22"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="brouwer.html#T3" target="_self" title="BROUWER:th.3">Th3</a></span>;<br/>
<a NAME="E17:22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A14</font></span>: 
<font color="Maroon" title="c4">o</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22"><span class="lab"><font color="Green" title="E21">A10</font></span></a>, <a class="ref" href="topreal9.html#T8" title="TOPREAL9:th.8">TOPREAL9:8</a></span>;<br/>
<a NAME="E18:22"/>
( <font color="Maroon" title="c14">s</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 <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span> &amp;  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal9.html#T17" title="TOPREAL9:th.17">TOPREAL9:17</a></span>;<br/>
<a NAME="E19:22"/><span class="kw">then </span>
<font color="Maroon" title="c14">s</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:22"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="ref" href="brouwer.html#T3" target="_self" title="BROUWER:th.3">Th3</a></span>;<br/>
<a NAME="E20:22"/><span class="kw">then </span>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c14">s</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:22"><span class="lab"><font color="Green" title="E24">A14</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E21:22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A15</font></span>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c14">s</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <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 <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span>,<span class="p2">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></span>;<br/>
<a NAME="E22:22"/>
<font color="Maroon" title="c14">s</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T55" title="PRE_TOPC:th.55">PRE_TOPC:55</a></span>;<br/>
<a NAME="E23:22"/><span class="kw">then </span>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c14">s</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="brouwer.html#K1" title="BROUWER:func.1">DiffElems</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:22"><span class="lab"><font color="Green" title="E23">A12</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">OK</font> = <span class="p1">(<span class="default"><a href="brouwer.html#K1" title="BROUWER:func.1">DiffElems</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span>,<span class="p2">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> 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 <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:22"><span class="lab"><font color="Green" title="E25">A15</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">Zf1</font> = <font color="Maroon" title="c10">f1</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>5</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  );<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>6</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">y1</font>, <font color="Olive" title="b2">y2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Olive" title="b1">y1</font>,<font color="Olive" title="b2">y2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">y1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> );<br/>
<span class="kw">set </span><font color="Maroon" title="c17">TD</font> = <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c15">OK</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c18">f</font> be   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span>,<span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">f</font> <a href="abian.html#NR4" title="ABIAN:NR.4">has_no_fixpoint</a>  implies  <a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c18">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E25:22"/><span class="lab"><font color="Green" title="E26">A16</font></span>: 
<font color="Maroon" title="c18">f</font> <a href="abian.html#NR4" title="ABIAN:NR.4">has_no_fixpoint</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c18">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> </span><br/>

<a NAME="E26:22"/><span class="lab"><font color="Green" title="E27">A17</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>  ex <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Maroon">S<sub>6</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">r</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Maroon">S<sub>6</sub></font>[<font color="Maroon" title="c19">x</font>,<font color="Olive" title="b1">r</font>]</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c20">x1</font>, <font color="Maroon" title="c21">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E2:22_6"/><span class="lab"><font color="Green" title="E28">A18</font></span>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c20">x1</font>,<font color="Maroon" title="c21">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T50" title="BORSUK_1:th.50">BORSUK_1:50</a></span>;<br/>
<span class="kw">take </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">x1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">x2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>6</sub></font>[<font color="Maroon" title="c19">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c20">x1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">x2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span>]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c20">x1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c20">x1</font>,<font color="Olive" title="b1">y2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c20">x1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">x2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">x1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">y2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c21">x2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c20">x1</font>,<font color="Maroon" title="c21">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c20">x1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">x2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">x1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">x2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E3:22_6"/>
( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c20">x1</font>,<font color="Maroon" title="c21">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c20">x1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">x2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">x1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">x2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_6"><span class="lab"><font color="Green" title="E28">A18</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c19">dy</font> being   <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E28:22"/><span class="lab"><font color="Green" title="E28">A19</font></span>: 
for <font color="Olive" title="b1">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> holds  <font color="Maroon">S<sub>6</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c19">dy</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E26:22"><span class="lab"><font color="Green" title="E27">A17</font></span></a>);<br/></span>
<a NAME="E29:22"/><span class="lab"><font color="Green" title="E29">A20</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>  ex <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">r</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c20">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c20">x</font>,<font color="Olive" title="b1">r</font>]</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c21">x1</font>, <font color="Maroon" title="c22">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E2:22_7"/><span class="lab"><font color="Green" title="E30">A21</font></span>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c21">x1</font>,<font color="Maroon" title="c22">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T50" title="BORSUK_1:th.50">BORSUK_1:50</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c22">x2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c22">x2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> ]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c21">x1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c21">x1</font>,<font color="Olive" title="b1">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> &amp; <font color="Maroon" title="c22">x2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c22">x2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c21">x1</font>,<font color="Maroon" title="c22">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> &amp; <font color="Maroon" title="c22">x2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">x2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E3:22_7"/>
( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c21">x1</font>,<font color="Maroon" title="c22">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> &amp; <font color="Maroon" title="c22">x2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">x2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_7"><span class="lab"><font color="Green" title="E30">A21</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c20">fy2</font> being   <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E31:22"/><span class="lab"><font color="Green" title="E30">A22</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> holds  <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c20">fy2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E29:22"><span class="lab"><font color="Green" title="E29">A20</font></span></a>);<br/></span>
<a NAME="E32:22"/><span class="lab"><font color="Green" title="E31">A23</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>  ex <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">r</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c21">x</font>,<font color="Olive" title="b1">r</font>]</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c22">x1</font>, <font color="Maroon" title="c23">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E2:22_8"/><span class="lab"><font color="Green" title="E32">A24</font></span>: 
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c22">x1</font>,<font color="Maroon" title="c23">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T50" title="BORSUK_1:th.50">BORSUK_1:50</a></span>;<br/>
<span class="kw">take </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c22">x1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">x2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c21">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">x1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">x2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span>]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c22">x1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c22">x1</font>,<font color="Olive" title="b1">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c22">x1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">x2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">x1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c23">x2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c22">x1</font>,<font color="Maroon" title="c23">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c22">x1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">x2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">x1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">x2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E3:22_8"/>
( <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c22">x1</font>,<font color="Maroon" title="c23">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c22">x1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">x2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">x1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">x2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_8"><span class="lab"><font color="Green" title="E32">A24</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c21">dx</font> being   <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E34:22"/><span class="lab"><font color="Green" title="E32">A25</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c21">dx</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E32:22"><span class="lab"><font color="Green" title="E31">A23</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c22">Dx</font> = <font color="Maroon" title="c21">dx</font>, <font color="Maroon" title="c23">Dy</font> = <font color="Maroon" title="c19">dy</font>, <font color="Maroon" title="c24">fX2</font> = <font color="Maroon" title="c8">fx2</font>, <font color="Maroon" title="c25">fY2</font> = <font color="Maroon" title="c20">fy2</font>, <font color="Maroon" title="c26">Xo</font> = <font color="Maroon" title="c7">xo</font>, <font color="Maroon" title="c27">Yo</font> = <font color="Maroon" title="c9">yo</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></span>;<br/>
<a NAME="E36:22"/>
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><br/> for <font color="Olive" title="b2">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Maroon" title="c27">Yo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">V</font> &amp; <font color="Olive" title="b2">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/> ex <font color="Olive" title="b3">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">W</font> &amp; <font color="Olive" title="b3">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c27">Yo</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b3">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c28">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Maroon" title="c27">Yo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b1">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/> ex <font color="Olive" title="b2">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Maroon" title="c28">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">W</font> &amp; <font color="Olive" title="b2">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c27">Yo</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">V</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c29">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c27">Yo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">V</font> &amp; <font color="Maroon" title="c29">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies  ex <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Maroon" title="c28">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c27">Yo</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c29">V</font> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:22_9"/><span class="lab"><font color="Green" title="E33">A26</font></span>: 
<font color="Maroon" title="c27">Yo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">V</font>
 <span class="kw">and </span><br/><a NAME="E2:22_9"/><span class="lab"><font color="Green" title="E34">A27</font></span>: 
<font color="Maroon" title="c29">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Maroon" title="c28">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c27">Yo</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c29">V</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c30">V1</font> = <font color="Maroon" title="c29">V</font> as   <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</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="#E2:22_9"><span class="lab"><font color="Green" title="E34">A27</font></span></a>, <a class="ref" href="borsuk_5.html#T62" title="BORSUK_5:th.62">BORSUK_5:62</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c31">p1</font>, <font color="Maroon" title="c32">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E5:22_9"/><span class="lab"><font color="Green" title="E35">A28</font></span>: 
<font color="Maroon" title="c28">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c31">p1</font>,<font color="Maroon" title="c32">p2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">and </span><br/><a NAME="E6:22_9"/><span class="lab"><font color="Green" title="E36">A29</font></span>: 
<font color="Maroon" title="c27">Yo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22"><span class="lab"><font color="Green" title="E20">A9</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c33">r</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c34">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> <span class="kw"> such that </span><br/><a NAME="E8:22_9"/><span class="lab"><font color="Green" title="E37">A30</font></span>: 
 <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="c34">g</font>
 <span class="kw">and </span><br/><a NAME="E9:22_9"/><span class="lab"><font color="Green" title="E38">A31</font></span>: 
<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"><span class="p4">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c34">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c34">g</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c30">V1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_9"><span class="lab"><font color="Green" title="E33">A26</font></span></a>, <a class="txt" href="#E6:22_9"><span class="lab"><font color="Green" title="E36">A29</font></span></a>, <a class="ref" href="rcomp_1.html#T40" title="RCOMP_1:th.40">RCOMP_1:40</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c35">g</font> = <font color="Maroon" title="c34">g</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" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c36">W2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c35">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c35">g</font> ) </span>}</span> ;<br/>
<a NAME="E11:22_9"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c35">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c35">g</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_9_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c37">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c37">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c35">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c35">g</font> ) </span>}</span>  or <font color="Maroon" title="c37">a</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_9_1"/>
<font color="Maroon" title="c37">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c35">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c35">g</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c37">a</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><br/>

<a NAME="E2:22_9_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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 <br/>( <font color="Maroon" title="c37">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c35">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c35">g</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:22_9_1"/>
<font color="Maroon" title="c37">a</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c37">W2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c35">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c35">g</font> ) </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<span class="kw">take </span>
<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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c37">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c28">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c37">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> &amp; <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c37">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c27">Yo</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c37">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c29">V</font> )</span><br/>

<a NAME="E13:22_9"/><span class="lab"><font color="Green" title="E39">A32</font></span>: 
<font color="Maroon" title="c32">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></span>;<br/>
<a NAME="E14:22_9"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c35">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c35">g</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_9"><span class="lab"><font color="Green" title="E37">A30</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></span>;<br/>
<a NAME="E15:22_9"/><span class="kw">then </span>
<font color="Maroon" title="c32">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c37">W2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22_9"><span class="lab"><font color="Green" title="E39">A32</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:22_9"/>
<font color="Maroon" title="c28">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c37">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_9"><span class="lab"><font color="Green" title="E35">A28</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c37">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c27">Yo</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c37">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c29">V</font> )</span><br/>

<a NAME="E17:22_9"/>
<font color="Maroon" title="c37">W2</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="pscomp_1.html#T68" title="PSCOMP_1:th.68">PSCOMP_1:68</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:22_9"/>
<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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c37">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> 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="borsuk_1.html#T46" title="BORSUK_1:th.46">BORSUK_1:46</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c27">Yo</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c37">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c29">V</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c38">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c38">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">Yo</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c37">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> or <font color="Maroon" title="c38">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">V</font> )</span><br/>

<span class="kw">assume </span><a NAME="E19:22_9"/>
<font color="Maroon" title="c38">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">Yo</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c37">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c38">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">V</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c39">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E21:22_9"/><span class="lab"><font color="Green" title="E40">A33</font></span>: 
<font color="Maroon" title="c39">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c37">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E22:22_9"/><span class="lab"><font color="Green" title="E41">A34</font></span>: 
<font color="Maroon" title="c27">Yo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c39">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c40">a1</font>, <font color="Maroon" title="c41">a2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E24:22_9"/><span class="lab"><font color="Green" title="E42">A35</font></span>: 
<font color="Maroon" title="c39">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c40">a1</font>,<font color="Maroon" title="c41">a2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">and </span><br/><a NAME="E25:22_9"/><span class="lab"><font color="Green" title="E43">A36</font></span>: 
<font color="Maroon" title="c9">yo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c39">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">a2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22"><span class="lab"><font color="Green" title="E20">A9</font></span></a></span>;<br/>
<a NAME="E26:22_9"/>
<font color="Maroon" title="c41">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c37">W2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:22_9"><span class="lab"><font color="Green" title="E40">A33</font></span></a>, <a class="txt" href="#E24:22_9"><span class="lab"><font color="Green" title="E42">A35</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c42">x2</font>, <font color="Maroon" title="c43">y2</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> <span class="kw"> such that </span><br/><a NAME="E28:22_9"/><span class="lab"><font color="Green" title="E44">A37</font></span>: 
<font color="Maroon" title="c41">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c42">x2</font>,<font color="Maroon" title="c43">y2</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">and </span><br/><a NAME="E29:22_9"/><span class="lab"><font color="Green" title="E45">A38</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c35">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c43">y2</font> &amp; <font color="Maroon" title="c43">y2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c35">g</font> )
 ;<br/>
<a NAME="E30:22_9"/>
<font color="Maroon" title="c41">a2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c43">y2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:22_9"><span class="lab"><font color="Green" title="E44">A37</font></span></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></span>;<br/>
<a NAME="E31:22_9"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c35">g</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </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="c41">a2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c41">a2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c35">g</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:22_9"><span class="lab"><font color="Green" title="E45">A38</font></span></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/>
<a NAME="E32:22_9"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c41">a2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c35">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c35">g</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E33:22_9"/>
<font color="Maroon" title="c38">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_9"><span class="lab"><font color="Green" title="E38">A31</font></span></a>, <a class="txt" href="#E22:22_9"><span class="lab"><font color="Green" title="E41">A34</font></span></a>, <a class="txt" href="#E25:22_9"><span class="lab"><font color="Green" title="E43">A36</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E37:22"/><span class="kw">then </span>
<font color="Maroon" title="c27">Yo</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c28">yo</font> = <font color="Maroon" title="c9">yo</font> as   <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a>  <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal6.html#T83" title="TOPREAL6:th.83">TOPREAL6:83</a></span>;<br/>
<a NAME="E39:22"/>
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><br/> for <font color="Olive" title="b2">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Maroon" title="c26">Xo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">V</font> &amp; <font color="Olive" title="b2">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/> ex <font color="Olive" title="b3">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">W</font> &amp; <font color="Olive" title="b3">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c26">Xo</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b3">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c29">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Maroon" title="c26">Xo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c29">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b1">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/> ex <font color="Olive" title="b2">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Maroon" title="c29">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">W</font> &amp; <font color="Olive" title="b2">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c26">Xo</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">V</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c30">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c26">Xo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c29">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c30">V</font> &amp; <font color="Maroon" title="c30">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies  ex <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Maroon" title="c29">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c26">Xo</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c30">V</font> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:22_10"/><span class="lab"><font color="Green" title="E33">A39</font></span>: 
<font color="Maroon" title="c26">Xo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c29">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c30">V</font>
 <span class="kw">and </span><br/><a NAME="E2:22_10"/><span class="lab"><font color="Green" title="E34">A40</font></span>: 
<font color="Maroon" title="c30">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Maroon" title="c29">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c26">Xo</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c30">V</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c31">V1</font> = <font color="Maroon" title="c30">V</font> as   <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</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="#E2:22_10"><span class="lab"><font color="Green" title="E34">A40</font></span></a>, <a class="ref" href="borsuk_5.html#T62" title="BORSUK_5:th.62">BORSUK_5:62</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c32">p1</font>, <font color="Maroon" title="c33">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E5:22_10"/><span class="lab"><font color="Green" title="E35">A41</font></span>: 
<font color="Maroon" title="c29">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c32">p1</font>,<font color="Maroon" title="c33">p2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">and </span><br/><a NAME="E6:22_10"/><span class="lab"><font color="Green" title="E36">A42</font></span>: 
<font color="Maroon" title="c26">Xo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c29">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22"><span class="lab"><font color="Green" title="E16">A3</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c34">r</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c33">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c35">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> <span class="kw"> such that </span><br/><a NAME="E8:22_10"/><span class="lab"><font color="Green" title="E37">A43</font></span>: 
 <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="c35">g</font>
 <span class="kw">and </span><br/><a NAME="E9:22_10"/><span class="lab"><font color="Green" title="E38">A44</font></span>: 
<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"><span class="p4">(<span class="default"><font color="Maroon" title="c33">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c35">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c33">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c35">g</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c31">V1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_10"><span class="lab"><font color="Green" title="E33">A39</font></span></a>, <a class="txt" href="#E6:22_10"><span class="lab"><font color="Green" title="E36">A42</font></span></a>, <a class="ref" href="rcomp_1.html#T40" title="RCOMP_1:th.40">RCOMP_1:40</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c36">g</font> = <font color="Maroon" title="c35">g</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" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c37">W2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c33">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c36">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c36">g</font> ) </span>}</span> ;<br/>
<a NAME="E11:22_10"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c33">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c36">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c36">g</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_10_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c38">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c38">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c33">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c36">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c36">g</font> ) </span>}</span>  or <font color="Maroon" title="c38">a</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_10_1"/>
<font color="Maroon" title="c38">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c33">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c36">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c36">g</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c38">a</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><br/>

<a NAME="E2:22_10_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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 <br/>( <font color="Maroon" title="c38">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c33">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c36">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c36">g</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:22_10_1"/>
<font color="Maroon" title="c38">a</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c38">W2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c33">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c36">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c36">g</font> ) </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<span class="kw">take </span>
<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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c38">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c29">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c38">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> &amp; <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c38">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c26">Xo</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c38">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c30">V</font> )</span><br/>

<a NAME="E13:22_10"/><span class="lab"><font color="Green" title="E39">A45</font></span>: 
<font color="Maroon" title="c33">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c33">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c33">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></span>;<br/>
<a NAME="E14:22_10"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c33">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c36">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c33">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c36">g</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_10"><span class="lab"><font color="Green" title="E37">A43</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></span>;<br/>
<a NAME="E15:22_10"/><span class="kw">then </span>
<font color="Maroon" title="c33">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c38">W2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22_10"><span class="lab"><font color="Green" title="E39">A45</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:22_10"/>
<font color="Maroon" title="c29">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c38">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_10"><span class="lab"><font color="Green" title="E35">A41</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c38">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c26">Xo</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c38">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c30">V</font> )</span><br/>

<a NAME="E17:22_10"/>
<font color="Maroon" title="c38">W2</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="pscomp_1.html#T66" title="PSCOMP_1:th.66">PSCOMP_1:66</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:22_10"/>
<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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c38">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> 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="borsuk_1.html#T46" title="BORSUK_1:th.46">BORSUK_1:46</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">Xo</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c38">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c30">V</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c39">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c39">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">Xo</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c38">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> or <font color="Maroon" title="c39">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c30">V</font> )</span><br/>

<span class="kw">assume </span><a NAME="E19:22_10"/>
<font color="Maroon" title="c39">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">Xo</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c38">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c39">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c30">V</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c40">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E21:22_10"/><span class="lab"><font color="Green" title="E40">A46</font></span>: 
<font color="Maroon" title="c40">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c38">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E22:22_10"/><span class="lab"><font color="Green" title="E41">A47</font></span>: 
<font color="Maroon" title="c26">Xo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c40">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c41">a1</font>, <font color="Maroon" title="c42">a2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E24:22_10"/><span class="lab"><font color="Green" title="E42">A48</font></span>: 
<font color="Maroon" title="c40">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c41">a1</font>,<font color="Maroon" title="c42">a2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">and </span><br/><a NAME="E25:22_10"/><span class="lab"><font color="Green" title="E43">A49</font></span>: 
<font color="Maroon" title="c7">xo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c40">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">a2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22"><span class="lab"><font color="Green" title="E16">A3</font></span></a></span>;<br/>
<a NAME="E26:22_10"/>
<font color="Maroon" title="c42">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c38">W2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:22_10"><span class="lab"><font color="Green" title="E40">A46</font></span></a>, <a class="txt" href="#E24:22_10"><span class="lab"><font color="Green" title="E42">A48</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c43">x2</font>, <font color="Maroon" title="c44">y2</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> <span class="kw"> such that </span><br/><a NAME="E28:22_10"/><span class="lab"><font color="Green" title="E44">A50</font></span>: 
<font color="Maroon" title="c42">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c43">x2</font>,<font color="Maroon" title="c44">y2</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">and </span><br/><a NAME="E29:22_10"/><span class="lab"><font color="Green" title="E45">A51</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c33">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c36">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c43">x2</font> &amp; <font color="Maroon" title="c43">x2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c36">g</font> )
 ;<br/>
<a NAME="E30:22_10"/>
<font color="Maroon" title="c42">a2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c43">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:22_10"><span class="lab"><font color="Green" title="E44">A50</font></span></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></span>;<br/>
<a NAME="E31:22_10"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c33">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c36">g</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </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="c42">a2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c42">a2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c33">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c36">g</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:22_10"><span class="lab"><font color="Green" title="E45">A51</font></span></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/>
<a NAME="E32:22_10"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c42">a2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c33">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c36">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c33">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c36">g</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E33:22_10"/>
<font color="Maroon" title="c39">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c30">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_10"><span class="lab"><font color="Green" title="E38">A44</font></span></a>, <a class="txt" href="#E22:22_10"><span class="lab"><font color="Green" title="E41">A47</font></span></a>, <a class="txt" href="#E25:22_10"><span class="lab"><font color="Green" title="E43">A49</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E40:22"/><span class="kw">then </span>
<font color="Maroon" title="c26">Xo</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c29">xo</font> = <font color="Maroon" title="c7">xo</font> as   <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a>  <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal6.html#T83" title="TOPREAL6:th.83">TOPREAL6:83</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c30">Zyo</font> = <font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c31">Zxo</font> = <font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c32">p2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K7" title="PARTFUN3:func.7">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c33">g</font> =  <a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c18">f</font>;<br/>
<a NAME="E42:22"/><span class="lab"><font color="Green" title="E33">A52</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">OK</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></span>;<br/>
<a NAME="E43:22"/>
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><br/> for <font color="Olive" title="b2">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Maroon" title="c23">Dy</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">V</font> &amp; <font color="Olive" title="b2">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/> ex <font color="Olive" title="b3">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">W</font> &amp; <font color="Olive" title="b3">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c23">Dy</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b3">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_11"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c34">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Maroon" title="c23">Dy</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c34">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b1">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/> ex <font color="Olive" title="b2">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Maroon" title="c34">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">W</font> &amp; <font color="Olive" title="b2">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c23">Dy</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">V</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c35">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">Dy</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c34">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c35">V</font> &amp; <font color="Maroon" title="c35">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies  ex <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Maroon" title="c34">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c23">Dy</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c35">V</font> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:22_11"/><span class="lab"><font color="Green" title="E34">A53</font></span>: 
<font color="Maroon" title="c23">Dy</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c34">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c35">V</font>
 <span class="kw">and </span><br/><a NAME="E2:22_11"/><span class="lab"><font color="Green" title="E35">A54</font></span>: 
<font color="Maroon" title="c35">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Maroon" title="c34">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c23">Dy</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c35">V</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c36">V1</font> = <font color="Maroon" title="c35">V</font> as   <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</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="#E2:22_11"><span class="lab"><font color="Green" title="E35">A54</font></span></a>, <a class="ref" href="borsuk_5.html#T62" title="BORSUK_5:th.62">BORSUK_5:62</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c37">p1</font>, <font color="Maroon" title="c38">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E5:22_11"/><span class="lab"><font color="Green" title="E36">A55</font></span>: 
<font color="Maroon" title="c34">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c37">p1</font>,<font color="Maroon" title="c38">p2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">and </span><br/><a NAME="E6:22_11"/><span class="lab"><font color="Green" title="E37">A56</font></span>: 
<font color="Maroon" title="c19">dy</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c34">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:22"><span class="lab"><font color="Green" title="E28">A19</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c39">r</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c40">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> <span class="kw"> such that </span><br/><a NAME="E8:22_11"/><span class="lab"><font color="Green" title="E38">A57</font></span>: 
 <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="c40">g</font>
 <span class="kw">and </span><br/><a NAME="E9:22_11"/><span class="lab"><font color="Green" title="E39">A58</font></span>: 
<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"><span class="p4">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c40">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c40">g</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c36">V1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_11"><span class="lab"><font color="Green" title="E34">A53</font></span></a>, <a class="txt" href="#E6:22_11"><span class="lab"><font color="Green" title="E37">A56</font></span></a>, <a class="ref" href="rcomp_1.html#T40" title="RCOMP_1:th.40">RCOMP_1:40</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c41">g</font> = <font color="Maroon" title="c40">g</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" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c42">W2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> ) </span>}</span> ;<br/>
<a NAME="E11:22_11"/><span class="lab"><font color="Green" title="E40">A59</font></span>: 
<span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_11_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c43">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c43">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> ) </span>}</span>  or <font color="Maroon" title="c43">a</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_11_1"/>
<font color="Maroon" title="c43">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c43">a</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><br/>

<a NAME="E2:22_11_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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 <br/>( <font color="Maroon" title="c43">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:22_11_1"/>
<font color="Maroon" title="c43">a</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E12:22_11"/><span class="lab"><font color="Green" title="E41">A60</font></span>: 
<font color="Maroon" title="c38">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c43">W2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> ) </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22_11"><span class="lab"><font color="Green" title="E40">A59</font></span></a></span>;<br/>
<a NAME="E14:22_11"/><span class="lab"><font color="Green" title="E42">A61</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_11"><span class="lab"><font color="Green" title="E38">A57</font></span></a>, <a class="ref" href="xreal_1.html#T76" title="XREAL_1:th.76">XREAL_1:76</a></span>;<br/>
<a NAME="E15:22_11"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</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="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></span>;<br/>
<a NAME="E16:22_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A62</font></span>: 
<font color="Maroon" title="c38">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c43">W2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:22_11"><span class="lab"><font color="Green" title="E41">A60</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c44">W1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> ) </span>}</span> ;<br/>
<a NAME="E17:22_11"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_11_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c45">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c45">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> ) </span>}</span>  or <font color="Maroon" title="c45">a</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_11_2"/>
<font color="Maroon" title="c45">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c45">a</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><br/>

<a NAME="E2:22_11_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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 <br/>( <font color="Maroon" title="c45">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:22_11_2"/>
<font color="Maroon" title="c45">a</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c45">W1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> ) </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<span class="kw">take </span>
<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c45">W1</font>,<font color="Maroon" title="c43">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c34">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c45">W1</font>,<font color="Maroon" title="c43">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> &amp; <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c45">W1</font>,<font color="Maroon" title="c43">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c23">Dy</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c45">W1</font>,<font color="Maroon" title="c43">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c35">V</font> )</span><br/>

<a NAME="E19:22_11"/><span class="lab"><font color="Green" title="E44">A63</font></span>: 
<font color="Maroon" title="c37">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></span>;<br/>
<a NAME="E20:22_11"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</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="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:22_11"><span class="lab"><font color="Green" title="E42">A61</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></span>;<br/>
<a NAME="E21:22_11"/><span class="kw">then </span>
<font color="Maroon" title="c37">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c45">W1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:22_11"><span class="lab"><font color="Green" title="E44">A63</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E22:22_11"/>
<font color="Maroon" title="c34">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c45">W1</font>,<font color="Maroon" title="c43">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_11"><span class="lab"><font color="Green" title="E36">A55</font></span></a>, <a class="txt" href="#E16:22_11"><span class="lab"><font color="Green" title="E43">A62</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c45">W1</font>,<font color="Maroon" title="c43">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c23">Dy</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c45">W1</font>,<font color="Maroon" title="c43">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c35">V</font> )</span><br/>

<a NAME="E23:22_11"/>
( <font color="Maroon" title="c45">W1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c43">W2</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="pscomp_1.html#T68" title="PSCOMP_1:th.68">PSCOMP_1:68</a></span>;<br/>
<span class="kw">hence </span><a NAME="E24:22_11"/>
<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c45">W1</font>,<font color="Maroon" title="c43">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> 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="borsuk_1.html#T46" title="BORSUK_1:th.46">BORSUK_1:46</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">Dy</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c45">W1</font>,<font color="Maroon" title="c43">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c35">V</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c46">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c46">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">Dy</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c45">W1</font>,<font color="Maroon" title="c43">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> or <font color="Maroon" title="c46">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c35">V</font> )</span><br/>

<span class="kw">assume </span><a NAME="E25:22_11"/>
<font color="Maroon" title="c46">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">Dy</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c45">W1</font>,<font color="Maroon" title="c43">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c46">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c35">V</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c47">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E27:22_11"/><span class="lab"><font color="Green" title="E45">A64</font></span>: 
<font color="Maroon" title="c47">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c45">W1</font>,<font color="Maroon" title="c43">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E28:22_11"/><span class="lab"><font color="Green" title="E46">A65</font></span>: 
<font color="Maroon" title="c23">Dy</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c47">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c46">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c48">a1</font>, <font color="Maroon" title="c49">a2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E30:22_11"/><span class="lab"><font color="Green" title="E47">A66</font></span>: 
<font color="Maroon" title="c47">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">a1</font>,<font color="Maroon" title="c49">a2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">and </span><br/><a NAME="E31:22_11"/><span class="lab"><font color="Green" title="E48">A67</font></span>: 
<font color="Maroon" title="c19">dy</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c47">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c48">a1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">a2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:22"><span class="lab"><font color="Green" title="E28">A19</font></span></a></span>;<br/>
<a NAME="E32:22_11"/>
<font color="Maroon" title="c49">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c43">W2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:22_11"><span class="lab"><font color="Green" title="E45">A64</font></span></a>, <a class="txt" href="#E30:22_11"><span class="lab"><font color="Green" title="E47">A66</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c50">x2</font>, <font color="Maroon" title="c51">y2</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> <span class="kw"> such that </span><br/><a NAME="E34:22_11"/><span class="lab"><font color="Green" title="E49">A68</font></span>: 
<font color="Maroon" title="c49">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c50">x2</font>,<font color="Maroon" title="c51">y2</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">and </span><br/><a NAME="E35:22_11"/><span class="lab"><font color="Green" title="E50">A69</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c51">y2</font>
 <span class="kw">and </span><br/><a NAME="E36:22_11"/><span class="lab"><font color="Green" title="E51">A70</font></span>: 
<font color="Maroon" title="c51">y2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 ;<br/>
<a NAME="E37:22_11"/><span class="lab"><font color="Green" title="E52">A71</font></span>: 
<font color="Maroon" title="c49">a2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c51">y2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:22_11"><span class="lab"><font color="Green" title="E49">A68</font></span></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></span>;<br/>
<a NAME="E38:22_11"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c51">y2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:22_11"><span class="lab"><font color="Green" title="E51">A70</font></span></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></span>;<br/>
<a NAME="E39:22_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A72</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c51">y2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
;<br/>
<a NAME="E40:22_11"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c51">y2</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:22_11"><span class="lab"><font color="Green" title="E50">A69</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E41:22_11"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c51">y2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c51">y2</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c51">y2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/>
<a NAME="E42:22_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A73</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c51">y2</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:22_11"><span class="lab"><font color="Green" title="E53">A72</font></span></a>, <a class="ref" href="seq_2.html#T9" title="SEQ_2:th.9">SEQ_2:9</a></span>;<br/>
<a NAME="E43:22_11"/>
<font color="Maroon" title="c48">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c45">W1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:22_11"><span class="lab"><font color="Green" title="E45">A64</font></span></a>, <a class="txt" href="#E30:22_11"><span class="lab"><font color="Green" title="E47">A66</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c52">x1</font>, <font color="Maroon" title="c53">y1</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> <span class="kw"> such that </span><br/><a NAME="E45:22_11"/><span class="lab"><font color="Green" title="E55">A74</font></span>: 
<font color="Maroon" title="c48">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c52">x1</font>,<font color="Maroon" title="c53">y1</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">and </span><br/><a NAME="E46:22_11"/><span class="lab"><font color="Green" title="E56">A75</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c53">y1</font>
 <span class="kw">and </span><br/><a NAME="E47:22_11"/><span class="lab"><font color="Green" title="E57">A76</font></span>: 
<font color="Maroon" title="c53">y1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 ;<br/>
<a NAME="E48:22_11"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c53">y1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:22_11"><span class="lab"><font color="Green" title="E57">A76</font></span></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></span>;<br/>
<a NAME="E49:22_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A77</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c53">y1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
;<br/>
<a NAME="E50:22_11"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c53">y1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c51">y2</font></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"><span class="p3">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c53">y1</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"><span class="p3">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c51">y2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T143" title="COMPLEX1:th.143">COMPLEX1:143</a></span>;<br/>
<a NAME="E51:22_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A78</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c53">y1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c51">y2</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"><span class="p3">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c53">y1</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"><span class="p3">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c51">y2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></span>;<br/>
<a NAME="E52:22_11"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c53">y1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:22_11"><span class="lab"><font color="Green" title="E56">A75</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E53:22_11"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c53">y1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c53">y1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c53">y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/>
<a NAME="E54:22_11"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c53">y1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:22_11"><span class="lab"><font color="Green" title="E58">A77</font></span></a>, <a class="ref" href="seq_2.html#T9" title="SEQ_2:th.9">SEQ_2:9</a></span>;<br/>
<a NAME="E55:22_11"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c53">y1</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"><span class="p3">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c51">y2</font></span>)</span></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="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</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="c41">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:22_11"><span class="lab"><font color="Green" title="E54">A73</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/>
<a NAME="E56:22_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A79</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c53">y1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c51">y2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c41">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:22_11"><span class="lab"><font color="Green" title="E59">A78</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E57:22_11"/>
<font color="Maroon" title="c48">a1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c53">y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:22_11"><span class="lab"><font color="Green" title="E55">A74</font></span></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></span>;<br/>
<a NAME="E58:22_11"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c48">a1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">a2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c41">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c37">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c38">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c41">g</font></span>)</span></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="#E37:22_11"><span class="lab"><font color="Green" title="E52">A71</font></span></a>, <a class="txt" href="#E56:22_11"><span class="lab"><font color="Green" title="E60">A79</font></span></a>, <a class="ref" href="rcomp_1.html#T8" title="RCOMP_1:th.8">RCOMP_1:8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E59:22_11"/>
<font color="Maroon" title="c46">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c35">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_11"><span class="lab"><font color="Green" title="E39">A58</font></span></a>, <a class="txt" href="#E28:22_11"><span class="lab"><font color="Green" title="E46">A65</font></span></a>, <a class="txt" href="#E31:22_11"><span class="lab"><font color="Green" title="E48">A67</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E44:22"/><span class="kw">then </span>
<font color="Maroon" title="c23">Dy</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c34">dy</font> = <font color="Maroon" title="c19">dy</font> as   <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a>  <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal6.html#T83" title="TOPREAL6:th.83">TOPREAL6:83</a></span>;<br/>
<a NAME="E46:22"/>
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><br/> for <font color="Olive" title="b2">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Maroon" title="c22">Dx</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">V</font> &amp; <font color="Olive" title="b2">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/> ex <font color="Olive" title="b3">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">W</font> &amp; <font color="Olive" title="b3">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c22">Dx</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b3">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_12"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c35">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Maroon" title="c22">Dx</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c35">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b1">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/> ex <font color="Olive" title="b2">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Maroon" title="c35">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">W</font> &amp; <font color="Olive" title="b2">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c22">Dx</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">V</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c36">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">Dx</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c35">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c36">V</font> &amp; <font color="Maroon" title="c36">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies  ex <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Maroon" title="c35">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c22">Dx</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c36">V</font> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:22_12"/><span class="lab"><font color="Green" title="E34">A80</font></span>: 
<font color="Maroon" title="c22">Dx</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c35">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c36">V</font>
 <span class="kw">and </span><br/><a NAME="E2:22_12"/><span class="lab"><font color="Green" title="E35">A81</font></span>: 
<font color="Maroon" title="c36">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Maroon" title="c35">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c22">Dx</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c36">V</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c37">V1</font> = <font color="Maroon" title="c36">V</font> as   <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</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="#E2:22_12"><span class="lab"><font color="Green" title="E35">A81</font></span></a>, <a class="ref" href="borsuk_5.html#T62" title="BORSUK_5:th.62">BORSUK_5:62</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c38">p1</font>, <font color="Maroon" title="c39">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E5:22_12"/><span class="lab"><font color="Green" title="E36">A82</font></span>: 
<font color="Maroon" title="c35">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c38">p1</font>,<font color="Maroon" title="c39">p2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">and </span><br/><a NAME="E6:22_12"/><span class="lab"><font color="Green" title="E37">A83</font></span>: 
<font color="Maroon" title="c21">dx</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c35">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:22"><span class="lab"><font color="Green" title="E32">A25</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c40">r</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c41">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> <span class="kw"> such that </span><br/><a NAME="E8:22_12"/><span class="lab"><font color="Green" title="E38">A84</font></span>: 
 <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="c41">g</font>
 <span class="kw">and </span><br/><a NAME="E9:22_12"/><span class="lab"><font color="Green" title="E39">A85</font></span>: 
<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"><span class="p4">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c41">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c41">g</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c37">V1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_12"><span class="lab"><font color="Green" title="E34">A80</font></span></a>, <a class="txt" href="#E6:22_12"><span class="lab"><font color="Green" title="E37">A83</font></span></a>, <a class="ref" href="rcomp_1.html#T40" title="RCOMP_1:th.40">RCOMP_1:40</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c42">g</font> = <font color="Maroon" title="c41">g</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" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c43">W2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> ) </span>}</span> ;<br/>
<a NAME="E11:22_12"/><span class="lab"><font color="Green" title="E40">A86</font></span>: 
<span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_12_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c44">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c44">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> ) </span>}</span>  or <font color="Maroon" title="c44">a</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_12_1"/>
<font color="Maroon" title="c44">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c44">a</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><br/>

<a NAME="E2:22_12_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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 <br/>( <font color="Maroon" title="c44">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:22_12_1"/>
<font color="Maroon" title="c44">a</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E12:22_12"/><span class="lab"><font color="Green" title="E41">A87</font></span>: 
<font color="Maroon" title="c39">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c44">W2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> ) </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22_12"><span class="lab"><font color="Green" title="E40">A86</font></span></a></span>;<br/>
<a NAME="E14:22_12"/><span class="lab"><font color="Green" title="E42">A88</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_12"><span class="lab"><font color="Green" title="E38">A84</font></span></a>, <a class="ref" href="xreal_1.html#T76" title="XREAL_1:th.76">XREAL_1:76</a></span>;<br/>
<a NAME="E15:22_12"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</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="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></span>;<br/>
<a NAME="E16:22_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A89</font></span>: 
<font color="Maroon" title="c39">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c44">W2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:22_12"><span class="lab"><font color="Green" title="E41">A87</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c45">W1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> ) </span>}</span> ;<br/>
<a NAME="E17:22_12"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_12_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c46">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c46">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> ) </span>}</span>  or <font color="Maroon" title="c46">a</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_12_2"/>
<font color="Maroon" title="c46">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c46">a</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><br/>

<a NAME="E2:22_12_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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 <br/>( <font color="Maroon" title="c46">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:22_12_2"/>
<font color="Maroon" title="c46">a</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c46">W1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> ) </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<span class="kw">take </span>
<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c46">W1</font>,<font color="Maroon" title="c44">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c35">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c46">W1</font>,<font color="Maroon" title="c44">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> &amp; <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c46">W1</font>,<font color="Maroon" title="c44">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c22">Dx</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c46">W1</font>,<font color="Maroon" title="c44">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c36">V</font> )</span><br/>

<a NAME="E19:22_12"/><span class="lab"><font color="Green" title="E44">A90</font></span>: 
<font color="Maroon" title="c38">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></span>;<br/>
<a NAME="E20:22_12"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</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="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:22_12"><span class="lab"><font color="Green" title="E42">A88</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></span>;<br/>
<a NAME="E21:22_12"/><span class="kw">then </span>
<font color="Maroon" title="c38">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c46">W1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:22_12"><span class="lab"><font color="Green" title="E44">A90</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E22:22_12"/>
<font color="Maroon" title="c35">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c46">W1</font>,<font color="Maroon" title="c44">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_12"><span class="lab"><font color="Green" title="E36">A82</font></span></a>, <a class="txt" href="#E16:22_12"><span class="lab"><font color="Green" title="E43">A89</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c46">W1</font>,<font color="Maroon" title="c44">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c22">Dx</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c46">W1</font>,<font color="Maroon" title="c44">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c36">V</font> )</span><br/>

<a NAME="E23:22_12"/>
( <font color="Maroon" title="c46">W1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c44">W2</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="pscomp_1.html#T66" title="PSCOMP_1:th.66">PSCOMP_1:66</a></span>;<br/>
<span class="kw">hence </span><a NAME="E24:22_12"/>
<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c46">W1</font>,<font color="Maroon" title="c44">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> 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="borsuk_1.html#T46" title="BORSUK_1:th.46">BORSUK_1:46</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c22">Dx</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c46">W1</font>,<font color="Maroon" title="c44">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c36">V</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c47">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c47">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">Dx</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c46">W1</font>,<font color="Maroon" title="c44">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> or <font color="Maroon" title="c47">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c36">V</font> )</span><br/>

<span class="kw">assume </span><a NAME="E25:22_12"/>
<font color="Maroon" title="c47">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">Dx</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c46">W1</font>,<font color="Maroon" title="c44">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c47">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c36">V</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c48">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E27:22_12"/><span class="lab"><font color="Green" title="E45">A91</font></span>: 
<font color="Maroon" title="c48">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c46">W1</font>,<font color="Maroon" title="c44">W2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E28:22_12"/><span class="lab"><font color="Green" title="E46">A92</font></span>: 
<font color="Maroon" title="c22">Dx</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c48">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c47">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c49">a1</font>, <font color="Maroon" title="c50">a2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E30:22_12"/><span class="lab"><font color="Green" title="E47">A93</font></span>: 
<font color="Maroon" title="c48">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c49">a1</font>,<font color="Maroon" title="c50">a2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">and </span><br/><a NAME="E31:22_12"/><span class="lab"><font color="Green" title="E48">A94</font></span>: 
<font color="Maroon" title="c21">dx</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c48">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">a1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c50">a2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:22"><span class="lab"><font color="Green" title="E32">A25</font></span></a></span>;<br/>
<a NAME="E32:22_12"/>
<font color="Maroon" title="c50">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c44">W2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:22_12"><span class="lab"><font color="Green" title="E45">A91</font></span></a>, <a class="txt" href="#E30:22_12"><span class="lab"><font color="Green" title="E47">A93</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c51">x2</font>, <font color="Maroon" title="c52">y2</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> <span class="kw"> such that </span><br/><a NAME="E34:22_12"/><span class="lab"><font color="Green" title="E49">A95</font></span>: 
<font color="Maroon" title="c50">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c51">x2</font>,<font color="Maroon" title="c52">y2</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">and </span><br/><a NAME="E35:22_12"/><span class="lab"><font color="Green" title="E50">A96</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c51">x2</font>
 <span class="kw">and </span><br/><a NAME="E36:22_12"/><span class="lab"><font color="Green" title="E51">A97</font></span>: 
<font color="Maroon" title="c51">x2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 ;<br/>
<a NAME="E37:22_12"/><span class="lab"><font color="Green" title="E52">A98</font></span>: 
<font color="Maroon" title="c50">a2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c51">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:22_12"><span class="lab"><font color="Green" title="E49">A95</font></span></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></span>;<br/>
<a NAME="E38:22_12"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c51">x2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:22_12"><span class="lab"><font color="Green" title="E51">A97</font></span></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></span>;<br/>
<a NAME="E39:22_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A99</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c51">x2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
;<br/>
<a NAME="E40:22_12"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c51">x2</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:22_12"><span class="lab"><font color="Green" title="E50">A96</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E41:22_12"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c51">x2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c51">x2</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c51">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/>
<a NAME="E42:22_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A100</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c51">x2</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:22_12"><span class="lab"><font color="Green" title="E53">A99</font></span></a>, <a class="ref" href="seq_2.html#T9" title="SEQ_2:th.9">SEQ_2:9</a></span>;<br/>
<a NAME="E43:22_12"/>
<font color="Maroon" title="c49">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c46">W1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:22_12"><span class="lab"><font color="Green" title="E45">A91</font></span></a>, <a class="txt" href="#E30:22_12"><span class="lab"><font color="Green" title="E47">A93</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c53">x1</font>, <font color="Maroon" title="c54">y1</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> <span class="kw"> such that </span><br/><a NAME="E45:22_12"/><span class="lab"><font color="Green" title="E55">A101</font></span>: 
<font color="Maroon" title="c49">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c53">x1</font>,<font color="Maroon" title="c54">y1</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">and </span><br/><a NAME="E46:22_12"/><span class="lab"><font color="Green" title="E56">A102</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c53">x1</font>
 <span class="kw">and </span><br/><a NAME="E47:22_12"/><span class="lab"><font color="Green" title="E57">A103</font></span>: 
<font color="Maroon" title="c53">x1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 ;<br/>
<a NAME="E48:22_12"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c53">x1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:22_12"><span class="lab"><font color="Green" title="E57">A103</font></span></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></span>;<br/>
<a NAME="E49:22_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A104</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c53">x1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
;<br/>
<a NAME="E50:22_12"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c53">x1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c51">x2</font></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"><span class="p3">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c53">x1</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"><span class="p3">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c51">x2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T143" title="COMPLEX1:th.143">COMPLEX1:143</a></span>;<br/>
<a NAME="E51:22_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A105</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c53">x1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c51">x2</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"><span class="p3">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c53">x1</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"><span class="p3">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c51">x2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></span>;<br/>
<a NAME="E52:22_12"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c53">x1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:22_12"><span class="lab"><font color="Green" title="E56">A102</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E53:22_12"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c53">x1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c53">x1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c53">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/>
<a NAME="E54:22_12"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c53">x1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:22_12"><span class="lab"><font color="Green" title="E58">A104</font></span></a>, <a class="ref" href="seq_2.html#T9" title="SEQ_2:th.9">SEQ_2:9</a></span>;<br/>
<a NAME="E55:22_12"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c53">x1</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"><span class="p3">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c51">x2</font></span>)</span></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="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</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="c42">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:22_12"><span class="lab"><font color="Green" title="E54">A100</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/>
<a NAME="E56:22_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A106</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c53">x1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c51">x2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c42">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:22_12"><span class="lab"><font color="Green" title="E59">A105</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E57:22_12"/>
<font color="Maroon" title="c49">a1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c53">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:22_12"><span class="lab"><font color="Green" title="E55">A101</font></span></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></span>;<br/>
<a NAME="E58:22_12"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c49">a1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c50">a2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c42">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c38">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c39">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c42">g</font></span>)</span></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="#E37:22_12"><span class="lab"><font color="Green" title="E52">A98</font></span></a>, <a class="txt" href="#E56:22_12"><span class="lab"><font color="Green" title="E60">A106</font></span></a>, <a class="ref" href="rcomp_1.html#T8" title="RCOMP_1:th.8">RCOMP_1:8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E59:22_12"/>
<font color="Maroon" title="c47">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c36">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_12"><span class="lab"><font color="Green" title="E39">A85</font></span></a>, <a class="txt" href="#E28:22_12"><span class="lab"><font color="Green" title="E46">A92</font></span></a>, <a class="txt" href="#E31:22_12"><span class="lab"><font color="Green" title="E48">A94</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E47:22"/><span class="kw">then </span>
<font color="Maroon" title="c22">Dx</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c35">dx</font> = <font color="Maroon" title="c21">dx</font> as   <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a>  <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal6.html#T83" title="TOPREAL6:th.83">TOPREAL6:83</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c36">Zdy</font> = <font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c37">Zdx</font> = <font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c38">m</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span>;<br/>
<a NAME="E49:22"/>
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><br/> for <font color="Olive" title="b2">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Maroon" title="c25">fY2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">V</font> &amp; <font color="Olive" title="b2">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/> ex <font color="Olive" title="b3">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">W</font> &amp; <font color="Olive" title="b3">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c25">fY2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b3">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_13"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c39">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Maroon" title="c25">fY2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c39">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b1">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/> ex <font color="Olive" title="b2">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Maroon" title="c39">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">W</font> &amp; <font color="Olive" title="b2">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c25">fY2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">V</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c40">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c25">fY2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c39">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c40">V</font> &amp; <font color="Maroon" title="c40">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies  ex <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Maroon" title="c39">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c25">fY2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c40">V</font> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:22_13"/><span class="lab"><font color="Green" title="E34">A107</font></span>: 
<font color="Maroon" title="c25">fY2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c39">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c40">V</font>
 <span class="kw">and </span><br/><a NAME="E2:22_13"/><span class="lab"><font color="Green" title="E35">A108</font></span>: 
<font color="Maroon" title="c40">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Maroon" title="c39">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c25">fY2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c40">V</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c41">V1</font> = <font color="Maroon" title="c40">V</font> as   <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</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="#E2:22_13"><span class="lab"><font color="Green" title="E35">A108</font></span></a>, <a class="ref" href="borsuk_5.html#T62" title="BORSUK_5:th.62">BORSUK_5:62</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c42">p1</font>, <font color="Maroon" title="c43">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E5:22_13"/><span class="lab"><font color="Green" title="E36">A109</font></span>: 
<font color="Maroon" title="c39">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c42">p1</font>,<font color="Maroon" title="c43">p2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">and </span><br/><a NAME="E6:22_13"/><span class="lab"><font color="Green" title="E37">A110</font></span>: 
<font color="Maroon" title="c25">fY2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c39">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c43">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:22"><span class="lab"><font color="Green" title="E30">A22</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c44">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> <span class="kw"> such that </span><br/><a NAME="E8:22_13"/><span class="lab"><font color="Green" title="E38">A111</font></span>: 
 <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="c44">g</font>
 <span class="kw">and </span><br/><a NAME="E9:22_13"/><span class="lab"><font color="Green" title="E39">A112</font></span>: 
<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="c43">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c44">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c43">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c44">g</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c41">V1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_13"><span class="lab"><font color="Green" title="E34">A107</font></span></a>, <a class="txt" href="#E6:22_13"><span class="lab"><font color="Green" title="E37">A110</font></span></a>, <a class="ref" href="rcomp_1.html#T40" title="RCOMP_1:th.40">RCOMP_1:40</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c45">g</font> = <font color="Maroon" title="c44">g</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" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c46">W1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c43">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c45">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c43">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c45">g</font> ) </span>}</span> ;<br/>
<a NAME="E11:22_13"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c43">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c45">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c43">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c45">g</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_13_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c47">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c47">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c43">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c45">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c43">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c45">g</font> ) </span>}</span>  or <font color="Maroon" title="c47">a</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_13_1"/>
<font color="Maroon" title="c47">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c43">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c45">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c43">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c45">g</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c47">a</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><br/>

<a NAME="E2:22_13_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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 <br/>( <font color="Maroon" title="c47">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c43">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c45">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c43">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c45">g</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:22_13_1"/>
<font color="Maroon" title="c47">a</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c47">W1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c43">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c45">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c43">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c45">g</font> ) </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<span class="kw">take </span>
<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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c47">W1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c39">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c47">W1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> &amp; <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c47">W1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c25">fY2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c47">W1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c40">V</font> )</span><br/>

<a NAME="E13:22_13"/><span class="lab"><font color="Green" title="E40">A113</font></span>: 
<font color="Maroon" title="c43">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c43">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c43">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></span>;<br/>
<a NAME="E14:22_13"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c43">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c45">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c43">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c43">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c43">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c45">g</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_13"><span class="lab"><font color="Green" title="E38">A111</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></span>;<br/>
<a NAME="E15:22_13"/><span class="kw">then </span>
<font color="Maroon" title="c43">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c47">W1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22_13"><span class="lab"><font color="Green" title="E40">A113</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:22_13"/>
<font color="Maroon" title="c39">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c47">W1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_13"><span class="lab"><font color="Green" title="E36">A109</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c47">W1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c25">fY2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c47">W1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c40">V</font> )</span><br/>

<a NAME="E17:22_13"/>
<font color="Maroon" title="c47">W1</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="pscomp_1.html#T68" title="PSCOMP_1:th.68">PSCOMP_1:68</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:22_13"/>
<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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c47">W1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> 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="borsuk_1.html#T46" title="BORSUK_1:th.46">BORSUK_1:46</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">fY2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c47">W1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c40">V</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c48">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c48">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">fY2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c47">W1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> or <font color="Maroon" title="c48">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c40">V</font> )</span><br/>

<span class="kw">assume </span><a NAME="E19:22_13"/>
<font color="Maroon" title="c48">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">fY2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c47">W1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c48">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c40">V</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c49">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E21:22_13"/><span class="lab"><font color="Green" title="E41">A114</font></span>: 
<font color="Maroon" title="c49">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c47">W1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E22:22_13"/><span class="lab"><font color="Green" title="E42">A115</font></span>: 
<font color="Maroon" title="c25">fY2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c49">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c48">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c50">a1</font>, <font color="Maroon" title="c51">a2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E24:22_13"/><span class="lab"><font color="Green" title="E43">A116</font></span>: 
<font color="Maroon" title="c49">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c50">a1</font>,<font color="Maroon" title="c51">a2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">and </span><br/><a NAME="E25:22_13"/><span class="lab"><font color="Green" title="E44">A117</font></span>: 
<font color="Maroon" title="c25">fY2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c49">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c51">a2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:22"><span class="lab"><font color="Green" title="E30">A22</font></span></a></span>;<br/>
<a NAME="E26:22_13"/>
<font color="Maroon" title="c51">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c47">W1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:22_13"><span class="lab"><font color="Green" title="E41">A114</font></span></a>, <a class="txt" href="#E24:22_13"><span class="lab"><font color="Green" title="E43">A116</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c52">x1</font>, <font color="Maroon" title="c53">y1</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> <span class="kw"> such that </span><br/><a NAME="E28:22_13"/><span class="lab"><font color="Green" title="E45">A118</font></span>: 
<font color="Maroon" title="c51">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c52">x1</font>,<font color="Maroon" title="c53">y1</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">and </span><br/><a NAME="E29:22_13"/><span class="lab"><font color="Green" title="E46">A119</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c43">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c45">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c53">y1</font>
 <span class="kw">and </span><br/><a NAME="E30:22_13"/><span class="lab"><font color="Green" title="E47">A120</font></span>: 
<font color="Maroon" title="c53">y1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c43">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c45">g</font>
 ;<br/>
<a NAME="E31:22_13"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c43">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c53">y1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c43">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c43">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c45">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:22_13"><span class="lab"><font color="Green" title="E47">A120</font></span></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></span>;<br/>
<a NAME="E32:22_13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A121</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c43">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c53">y1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c45">g</font>
 
;<br/>
<a NAME="E33:22_13"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c43">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c45">g</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c45">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c53">y1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c45">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:22_13"><span class="lab"><font color="Green" title="E46">A119</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E34:22_13"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c43">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c53">y1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c53">y1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c45">g</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c53">y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/>
<a NAME="E35:22_13"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c43">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c53">y1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c45">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:22_13"><span class="lab"><font color="Green" title="E48">A121</font></span></a>, <a class="ref" href="seq_2.html#T9" title="SEQ_2:th.9">SEQ_2:9</a></span>;<br/>
<a NAME="E36:22_13"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c43">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c53">y1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c45">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></span>;<br/>
<a NAME="E37:22_13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A122</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c53">y1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c43">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c45">g</font>
 
;<br/>
<a NAME="E38:22_13"/>
<font color="Maroon" title="c51">a2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c53">y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:22_13"><span class="lab"><font color="Green" title="E45">A118</font></span></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></span>;<br/>
<a NAME="E39:22_13"/><span class="kw">then </span>
<font color="Maroon" title="c51">a2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <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"><span class="p3">(<span class="default"><font color="Maroon" title="c43">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c45">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c43">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c45">g</font></span>)</span></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="#E37:22_13"><span class="lab"><font color="Green" title="E49">A122</font></span></a>, <a class="ref" href="rcomp_1.html#T8" title="RCOMP_1:th.8">RCOMP_1:8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E40:22_13"/>
<font color="Maroon" title="c48">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c40">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_13"><span class="lab"><font color="Green" title="E39">A112</font></span></a>, <a class="txt" href="#E22:22_13"><span class="lab"><font color="Green" title="E42">A115</font></span></a>, <a class="txt" href="#E25:22_13"><span class="lab"><font color="Green" title="E44">A117</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E50:22"/><span class="kw">then </span>
<font color="Maroon" title="c25">fY2</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c39">fy2</font> = <font color="Maroon" title="c20">fy2</font> as   <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a>  <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal6.html#T83" title="TOPREAL6:th.83">TOPREAL6:83</a></span>;<br/>
<a NAME="E52:22"/>
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><br/> for <font color="Olive" title="b2">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Maroon" title="c24">fX2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">V</font> &amp; <font color="Olive" title="b2">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/> ex <font color="Olive" title="b3">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">W</font> &amp; <font color="Olive" title="b3">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c24">fX2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b3">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_14"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c40">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Maroon" title="c24">fX2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c40">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b1">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/> ex <font color="Olive" title="b2">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Maroon" title="c40">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">W</font> &amp; <font color="Olive" title="b2">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c24">fX2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">V</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c41">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">fX2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c40">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c41">V</font> &amp; <font color="Maroon" title="c41">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies  ex <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Maroon" title="c40">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c24">fX2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c41">V</font> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:22_14"/><span class="lab"><font color="Green" title="E34">A123</font></span>: 
<font color="Maroon" title="c24">fX2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c40">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c41">V</font>
 <span class="kw">and </span><br/><a NAME="E2:22_14"/><span class="lab"><font color="Green" title="E35">A124</font></span>: 
<font color="Maroon" title="c41">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Maroon" title="c40">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c24">fX2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c41">V</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c42">V1</font> = <font color="Maroon" title="c41">V</font> as   <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</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="#E2:22_14"><span class="lab"><font color="Green" title="E35">A124</font></span></a>, <a class="ref" href="borsuk_5.html#T62" title="BORSUK_5:th.62">BORSUK_5:62</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c43">p1</font>, <font color="Maroon" title="c44">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E5:22_14"/><span class="lab"><font color="Green" title="E36">A125</font></span>: 
<font color="Maroon" title="c40">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c43">p1</font>,<font color="Maroon" title="c44">p2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">and </span><br/><a NAME="E6:22_14"/><span class="lab"><font color="Green" title="E37">A126</font></span>: 
<font color="Maroon" title="c24">fX2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c40">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22"><span class="lab"><font color="Green" title="E18">A6</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c45">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> <span class="kw"> such that </span><br/><a NAME="E8:22_14"/><span class="lab"><font color="Green" title="E38">A127</font></span>: 
 <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="c45">g</font>
 <span class="kw">and </span><br/><a NAME="E9:22_14"/><span class="lab"><font color="Green" title="E39">A128</font></span>: 
<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="c44">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c45">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c44">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c45">g</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c42">V1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_14"><span class="lab"><font color="Green" title="E34">A123</font></span></a>, <a class="txt" href="#E6:22_14"><span class="lab"><font color="Green" title="E37">A126</font></span></a>, <a class="ref" href="rcomp_1.html#T40" title="RCOMP_1:th.40">RCOMP_1:40</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c46">g</font> = <font color="Maroon" title="c45">g</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" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c47">W1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c44">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c46">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c44">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c46">g</font> ) </span>}</span> ;<br/>
<a NAME="E11:22_14"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c44">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c46">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c44">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c46">g</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_14_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c48">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c48">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c44">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c46">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c44">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c46">g</font> ) </span>}</span>  or <font color="Maroon" title="c48">a</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_14_1"/>
<font color="Maroon" title="c48">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c44">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c46">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c44">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c46">g</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c48">a</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><br/>

<a NAME="E2:22_14_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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 <br/>( <font color="Maroon" title="c48">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c44">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c46">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c44">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c46">g</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:22_14_1"/>
<font color="Maroon" title="c48">a</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c48">W1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  : ( <span class="p1">(<span class="default"><font color="Maroon" title="c44">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c46">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c44">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c46">g</font> ) </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<span class="kw">take </span>
<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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c48">W1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c40">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c48">W1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> &amp; <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c48">W1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c24">fX2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c48">W1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c41">V</font> )</span><br/>

<a NAME="E13:22_14"/><span class="lab"><font color="Green" title="E40">A129</font></span>: 
<font color="Maroon" title="c44">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c44">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c44">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></span>;<br/>
<a NAME="E14:22_14"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c44">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c46">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c44">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c44">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c44">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c46">g</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_14"><span class="lab"><font color="Green" title="E38">A127</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></span>;<br/>
<a NAME="E15:22_14"/><span class="kw">then </span>
<font color="Maroon" title="c44">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c48">W1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22_14"><span class="lab"><font color="Green" title="E40">A129</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:22_14"/>
<font color="Maroon" title="c40">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c48">W1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_14"><span class="lab"><font color="Green" title="E36">A125</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c48">W1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c24">fX2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c48">W1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c41">V</font> )</span><br/>

<a NAME="E17:22_14"/>
<font color="Maroon" title="c48">W1</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="pscomp_1.html#T66" title="PSCOMP_1:th.66">PSCOMP_1:66</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:22_14"/>
<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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c48">W1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> 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="borsuk_1.html#T46" title="BORSUK_1:th.46">BORSUK_1:46</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">fX2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c48">W1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c41">V</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c49">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c49">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">fX2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c48">W1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> or <font color="Maroon" title="c49">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c41">V</font> )</span><br/>

<span class="kw">assume </span><a NAME="E19:22_14"/>
<font color="Maroon" title="c49">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">fX2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c48">W1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c49">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c41">V</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c50">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E21:22_14"/><span class="lab"><font color="Green" title="E41">A130</font></span>: 
<font color="Maroon" title="c50">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c48">W1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E22:22_14"/><span class="lab"><font color="Green" title="E42">A131</font></span>: 
<font color="Maroon" title="c24">fX2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c50">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c49">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c51">a1</font>, <font color="Maroon" title="c52">a2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E24:22_14"/><span class="lab"><font color="Green" title="E43">A132</font></span>: 
<font color="Maroon" title="c50">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c51">a1</font>,<font color="Maroon" title="c52">a2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">and </span><br/><a NAME="E25:22_14"/><span class="lab"><font color="Green" title="E44">A133</font></span>: 
<font color="Maroon" title="c24">fX2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c50">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c52">a2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22"><span class="lab"><font color="Green" title="E18">A6</font></span></a></span>;<br/>
<a NAME="E26:22_14"/>
<font color="Maroon" title="c52">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c48">W1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:22_14"><span class="lab"><font color="Green" title="E41">A130</font></span></a>, <a class="txt" href="#E24:22_14"><span class="lab"><font color="Green" title="E43">A132</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c53">x1</font>, <font color="Maroon" title="c54">y1</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> <span class="kw"> such that </span><br/><a NAME="E28:22_14"/><span class="lab"><font color="Green" title="E45">A134</font></span>: 
<font color="Maroon" title="c52">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c53">x1</font>,<font color="Maroon" title="c54">y1</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">and </span><br/><a NAME="E29:22_14"/><span class="lab"><font color="Green" title="E46">A135</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c44">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c46">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c53">x1</font>
 <span class="kw">and </span><br/><a NAME="E30:22_14"/><span class="lab"><font color="Green" title="E47">A136</font></span>: 
<font color="Maroon" title="c53">x1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c44">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c46">g</font>
 ;<br/>
<a NAME="E31:22_14"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c44">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c53">x1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c44">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c44">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c46">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:22_14"><span class="lab"><font color="Green" title="E47">A136</font></span></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></span>;<br/>
<a NAME="E32:22_14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A137</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c44">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c53">x1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c46">g</font>
 
;<br/>
<a NAME="E33:22_14"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c44">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c46">g</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c46">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c53">x1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c46">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:22_14"><span class="lab"><font color="Green" title="E46">A135</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E34:22_14"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c44">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c53">x1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c53">x1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c46">g</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c53">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/>
<a NAME="E35:22_14"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c44">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c53">x1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c46">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:22_14"><span class="lab"><font color="Green" title="E48">A137</font></span></a>, <a class="ref" href="seq_2.html#T9" title="SEQ_2:th.9">SEQ_2:9</a></span>;<br/>
<a NAME="E36:22_14"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c44">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c53">x1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c46">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></span>;<br/>
<a NAME="E37:22_14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A138</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c53">x1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c44">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c46">g</font>
 
;<br/>
<a NAME="E38:22_14"/>
<font color="Maroon" title="c52">a2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c53">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:22_14"><span class="lab"><font color="Green" title="E45">A134</font></span></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></span>;<br/>
<a NAME="E39:22_14"/><span class="kw">then </span>
<font color="Maroon" title="c52">a2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <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"><span class="p3">(<span class="default"><font color="Maroon" title="c44">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c46">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c44">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c46">g</font></span>)</span></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="#E37:22_14"><span class="lab"><font color="Green" title="E49">A138</font></span></a>, <a class="ref" href="rcomp_1.html#T8" title="RCOMP_1:th.8">RCOMP_1:8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E40:22_14"/>
<font color="Maroon" title="c49">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c41">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_14"><span class="lab"><font color="Green" title="E39">A128</font></span></a>, <a class="txt" href="#E22:22_14"><span class="lab"><font color="Green" title="E42">A131</font></span></a>, <a class="txt" href="#E25:22_14"><span class="lab"><font color="Green" title="E44">A133</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E53:22"/><span class="kw">then </span>
<font color="Maroon" title="c24">fX2</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c40">fx2</font> = <font color="Maroon" title="c8">fx2</font> as   <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a>  <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal6.html#T83" title="TOPREAL6:th.83">TOPREAL6:83</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c41">yy</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c42">xx</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c43">Zfy2</font> = <font color="Maroon" title="c39">fy2</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c44">Zfx2</font> = <font color="Maroon" title="c40">fx2</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c45">p1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E55:22"/><span class="lab"><font color="Green" title="E34">A139</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K7" title="PARTFUN3:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c15">OK</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E56:22"/><span class="lab"><font color="Green" title="E35">A140</font></span>: 
for <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">z</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">z</font> holds <br/><span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">OK</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_15"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c46">y</font>, <font color="Maroon" title="c47">z</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c46">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c47">z</font> implies <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c46">y</font>,<font color="Maroon" title="c47">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">OK</font> )</span><br/>

<a NAME="E1:22_15"/><span class="lab"><font color="Green" title="E36">A141</font></span>: 
( <font color="Maroon" title="c46">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> &amp; <font color="Maroon" title="c47">z</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T55" title="PRE_TOPC:th.55">PRE_TOPC:55</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:22_15"/>
<font color="Maroon" title="c46">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c47">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c46">y</font>,<font color="Maroon" title="c47">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">OK</font></span><br/>

<a NAME="E3:22_15"/><span class="kw">then </span>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c46">y</font>,<font color="Maroon" title="c47">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="brouwer.html#K1" title="BROUWER:func.1">DiffElems</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_15"><span class="lab"><font color="Green" title="E36">A141</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:22_15"/>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c46">y</font>,<font color="Maroon" title="c47">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">OK</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E36">A142</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_16"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c46">b</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c46">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K7" title="PARTFUN3:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> implies  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c46">b</font> )</span><br/><span class="kw">assume </span><a NAME="E1:22_16"/>
<font color="Maroon" title="c46">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K7" title="PARTFUN3:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c46">b</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c47">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:22_16"/><span class="lab"><font color="Green" title="E37">A143</font></span>: 
<font color="Maroon" title="c47">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K7" title="PARTFUN3:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:22_16"/><span class="lab"><font color="Green" title="E38">A144</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K7" title="PARTFUN3:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c46">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E5:22_16"/>
<font color="Maroon" title="c47">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="brouwer.html#K1" title="BROUWER:func.1">DiffElems</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:22"><span class="lab"><font color="Green" title="E33">A52</font></span></a>, <a class="txt" href="#E3:22_16"><span class="lab"><font color="Green" title="E37">A143</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c48">y</font>, <font color="Maroon" title="c49">z</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E7:22_16"/><span class="lab"><font color="Green" title="E39">A145</font></span>: 
<font color="Maroon" title="c47">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">and </span><br/><a NAME="E8:22_16"/><span class="lab"><font color="Green" title="E40">A146</font></span>: 
<font color="Maroon" title="c48">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c49">z</font>
 ;<br/><a NAME="E9:22_16"/>
<font color="Maroon" title="c47">a</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 <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span>,<span class="p2">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:22"><span class="lab"><font color="Green" title="E33">A52</font></span></a>, <a class="txt" href="#E3:22_16"><span class="lab"><font color="Green" title="E37">A143</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c50">a1</font>, <font color="Maroon" title="c51">a2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E11:22_16"/><span class="lab"><font color="Green" title="E41">A147</font></span>: 
<font color="Maroon" title="c47">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c50">a1</font>,<font color="Maroon" title="c51">a2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T50" title="BORSUK_1:th.50">BORSUK_1:50</a></span>;<br/><a NAME="E12:22_16"/><span class="lab"><font color="Green" title="E42">A148</font></span>: 
<font color="Maroon" title="c51">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c49">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_16"><span class="lab"><font color="Green" title="E39">A145</font></span></a>, <a class="txt" href="#E11:22_16"><span class="lab"><font color="Green" title="E41">A147</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E13:22_16"/><span class="lab"><font color="Green" title="E43">A149</font></span>: 
<font color="Maroon" title="c50">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c48">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_16"><span class="lab"><font color="Green" title="E39">A145</font></span></a>, <a class="txt" href="#E11:22_16"><span class="lab"><font color="Green" title="E41">A147</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E14:22_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A150</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:22"><span class="lab"><font color="Green" title="E35">A140</font></span></a>, <a class="txt" href="#E8:22_16"><span class="lab"><font color="Green" title="E40">A146</font></span></a>, <a class="txt" href="#E12:22_16"><span class="lab"><font color="Green" title="E42">A148</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c52">r3</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c49">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c53">r4</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c49">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span>;<br/><span class="kw">consider </span><font color="Maroon" title="c54">x9</font>, <font color="Maroon" title="c55">x10</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E16:22_16"/><span class="lab"><font color="Green" title="E45">A151</font></span>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c54">x9</font>,<font color="Maroon" title="c55">x10</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">and </span><br/><a NAME="E17:22_16"/><span class="lab"><font color="Green" title="E46">A152</font></span>: 
<font color="Maroon" title="c29">xo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c55">x10</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22"><span class="lab"><font color="Green" title="E16">A3</font></span></a></span>;<br/><a NAME="E18:22_16"/><span class="lab"><font color="Green" title="E47">A153</font></span>: 
<font color="Maroon" title="c49">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c55">x10</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:22_16"><span class="lab"><font color="Green" title="E45">A151</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E19:22_16"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="brouwer.html#T3" target="_self" title="BROUWER:th.3">Th3</a></span>;<br/><a NAME="E20:22_16"/><span class="kw">then </span>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c49">z</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">o</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:22_16"><span class="lab"><font color="Green" title="E42">A148</font></span></a>, <a class="ref" href="topreal9.html#T8" title="TOPREAL9:th.8">TOPREAL9:8</a></span>;<br/><a NAME="E21:22_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A154</font></span>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c49">z</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">o</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c56">x11</font>, <font color="Maroon" title="c57">x12</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E23:22_16"/><span class="lab"><font color="Green" title="E49">A155</font></span>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">x11</font>,<font color="Maroon" title="c57">x12</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">and </span><br/><a NAME="E24:22_16"/><span class="lab"><font color="Green" title="E50">A156</font></span>: 
<font color="Maroon" title="c28">yo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c57">x12</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22"><span class="lab"><font color="Green" title="E20">A9</font></span></a></span>;<br/><a NAME="E25:22_16"/><span class="lab"><font color="Green" title="E51">A157</font></span>: 
<font color="Maroon" title="c49">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c57">x12</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:22_16"><span class="lab"><font color="Green" title="E49">A155</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E26:22_16"/><span class="lab"><font color="Green" title="E52">A158</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">xo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">yo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:22"><span class="lab"><font color="Green" title="E35">A140</font></span></a>, <a class="txt" href="#E8:22_16"><span class="lab"><font color="Green" title="E40">A146</font></span></a>, <a class="txt" href="#E13:22_16"><span class="lab"><font color="Green" title="E43">A149</font></span></a>, <a class="txt" href="#E12:22_16"><span class="lab"><font color="Green" title="E42">A148</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c49">z</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">o</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c49">z</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">o</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c49">z</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">o</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_1.html#T46" title="JGRAPH_1:th.46">JGRAPH_1:46</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c49">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c49">z</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">o</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal3.html#T8" title="TOPREAL3:th.8">TOPREAL3:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c49">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c49">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal3.html#T8" title="TOPREAL3:th.8">TOPREAL3:8</a></span>
;<br/><a NAME="E28:22_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A159</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c49">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c49">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:22_16"><span class="lab"><font color="Green" title="E48">A154</font></span></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K7" title="PARTFUN3:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></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"><font color="Maroon" title="c10">f1</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_16"><span class="lab"><font color="Green" title="E37">A143</font></span></a>, <a class="txt" href="#E7:22_16"><span class="lab"><font color="Green" title="E39">A145</font></span></a>, <a class="ref" href="valued_1.html#T13" title="VALUED_1:th.13">VALUED_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:22_16"><span class="lab"><font color="Green" title="E44">A150</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_16"><span class="lab"><font color="Green" title="E37">A143</font></span></a>, <a class="txt" href="#E7:22_16"><span class="lab"><font color="Green" title="E39">A145</font></span></a>, <a class="ref" href="valued_1.html#T1" title="VALUED_1:th.1">VALUED_1:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c49">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c49">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:22_16"><span class="lab"><font color="Green" title="E52">A158</font></span></a>, <a class="txt" href="#E17:22_16"><span class="lab"><font color="Green" title="E46">A152</font></span></a>, <a class="txt" href="#E18:22_16"><span class="lab"><font color="Green" title="E47">A153</font></span></a>, <a class="txt" href="#E24:22_16"><span class="lab"><font color="Green" title="E50">A156</font></span></a>, <a class="txt" href="#E25:22_16"><span class="lab"><font color="Green" title="E51">A157</font></span></a>, <a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></span>
;<br/><span class="kw">hence </span><a NAME="E30:22_16"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c46">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_16"><span class="lab"><font color="Green" title="E38">A144</font></span></a>, <a class="txt" href="#E7:22_16"><span class="lab"><font color="Green" title="E39">A145</font></span></a>, <a class="txt" href="#E28:22_16"><span class="lab"><font color="Green" title="E53">A159</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_17"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c46">b</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c46">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> implies  <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="c46">b</font> )</span><br/><span class="kw">assume </span><a NAME="E1:22_17"/>
<font color="Maroon" title="c46">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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="c46">b</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c47">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:22_17"/><span class="lab"><font color="Green" title="E37">A160</font></span>: 
<font color="Maroon" title="c47">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:22_17"/><span class="lab"><font color="Green" title="E38">A161</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c46">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E5:22_17"/>
<font color="Maroon" title="c47">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="brouwer.html#K1" title="BROUWER:func.1">DiffElems</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:22"><span class="lab"><font color="Green" title="E33">A52</font></span></a>, <a class="txt" href="#E3:22_17"><span class="lab"><font color="Green" title="E37">A160</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c48">y</font>, <font color="Maroon" title="c49">z</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E7:22_17"/><span class="lab"><font color="Green" title="E39">A162</font></span>: 
<font color="Maroon" title="c47">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">and </span><br/><a NAME="E8:22_17"/><span class="lab"><font color="Green" title="E40">A163</font></span>: 
<font color="Maroon" title="c48">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c49">z</font>
 ;<br/><a NAME="E9:22_17"/>
<font color="Maroon" title="c47">a</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 <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span>,<span class="p2">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:22"><span class="lab"><font color="Green" title="E33">A52</font></span></a>, <a class="txt" href="#E3:22_17"><span class="lab"><font color="Green" title="E37">A160</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c50">a1</font>, <font color="Maroon" title="c51">a2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E11:22_17"/><span class="lab"><font color="Green" title="E41">A164</font></span>: 
<font color="Maroon" title="c47">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c50">a1</font>,<font color="Maroon" title="c51">a2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T50" title="BORSUK_1:th.50">BORSUK_1:50</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c52">r1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c48">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c53">r2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c48">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span>;<br/><div><span class="lab"><font color="Green" title="E42">A165</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_17_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:22_17_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c48">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c49">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c48">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c49">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:22_17_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c48">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c48">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T2" title="COMPLEX1:th.2">COMPLEX1:2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:22_17_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_17"><span class="lab"><font color="Green" title="E40">A163</font></span></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c54">x3</font>, <font color="Maroon" title="c55">x4</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E14:22_17"/><span class="lab"><font color="Green" title="E43">A166</font></span>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c54">x3</font>,<font color="Maroon" title="c55">x4</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">and </span><br/><a NAME="E15:22_17"/><span class="lab"><font color="Green" title="E44">A167</font></span>: 
<font color="Maroon" title="c35">dx</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">x3</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c55">x4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:22"><span class="lab"><font color="Green" title="E32">A25</font></span></a></span>;<br/><a NAME="E16:22_17"/><span class="lab"><font color="Green" title="E45">A168</font></span>: 
( <font color="Maroon" title="c48">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c54">x3</font> &amp; <font color="Maroon" title="c49">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c55">x4</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:22_17"><span class="lab"><font color="Green" title="E43">A166</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E17:22_17"/>
( <font color="Maroon" title="c50">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c48">y</font> &amp; <font color="Maroon" title="c51">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c49">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_17"><span class="lab"><font color="Green" title="E39">A162</font></span></a>, <a class="txt" href="#E11:22_17"><span class="lab"><font color="Green" title="E41">A164</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E18:22_17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A169</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">dx</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">dy</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:22"><span class="lab"><font color="Green" title="E35">A140</font></span></a>, <a class="txt" href="#E8:22_17"><span class="lab"><font color="Green" title="E40">A163</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c56">x5</font>, <font color="Maroon" title="c57">x6</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E20:22_17"/><span class="lab"><font color="Green" title="E47">A170</font></span>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">x5</font>,<font color="Maroon" title="c57">x6</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">and </span><br/><a NAME="E21:22_17"/><span class="lab"><font color="Green" title="E48">A171</font></span>: 
<font color="Maroon" title="c34">dy</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c56">x5</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c57">x6</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:22"><span class="lab"><font color="Green" title="E28">A19</font></span></a></span>;<br/><a NAME="E22:22_17"/><span class="lab"><font color="Green" title="E49">A172</font></span>: 
( <font color="Maroon" title="c48">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c56">x5</font> &amp; <font color="Maroon" title="c49">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c57">x6</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:22_17"><span class="lab"><font color="Green" title="E47">A170</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_17"><span class="lab"><font color="Green" title="E37">A160</font></span></a>, <a class="txt" href="#E7:22_17"><span class="lab"><font color="Green" title="E39">A162</font></span></a>, <a class="ref" href="valued_1.html#T1" title="VALUED_1:th.1">VALUED_1:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c48">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c49">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c48">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c49">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:22_17"><span class="lab"><font color="Green" title="E44">A167</font></span></a>, <a class="txt" href="#E16:22_17"><span class="lab"><font color="Green" title="E45">A168</font></span></a>, <a class="txt" href="#E21:22_17"><span class="lab"><font color="Green" title="E48">A171</font></span></a>, <a class="txt" href="#E22:22_17"><span class="lab"><font color="Green" title="E49">A172</font></span></a>, <a class="txt" href="#E18:22_17"><span class="lab"><font color="Green" title="E46">A169</font></span></a>, <a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></span>
;<br/><span class="kw">hence </span><a NAME="E24:22_17"/>
 <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="c46">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_17"><span class="lab"><font color="Green" title="E38">A161</font></span></a>, <a class="txt" href="#E7:22_17"><span class="lab"><font color="Green" title="E39">A162</font></span></a>, <a class="txt" href="#E12:22_17"><span class="lab"><font color="Green" title="E42">A165</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c46">m</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> as   <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a>   <a href="partfun3.html#V1" title="PARTFUN3:attr.1">positive-yielding</a>  <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun3.html#D1" title="PARTFUN3:def.1">PARTFUN3:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c47">p2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K7" title="PARTFUN3:func.7">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> as   <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a>   <a href="partfun3.html#V3" title="PARTFUN3:attr.3">nonpositive-yielding</a>  <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:22"><span class="lab"><font color="Green" title="E36">A142</font></span></a>, <a class="ref" href="partfun3.html#D3" title="PARTFUN3:def.3">PARTFUN3:def 3</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c48">pp</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K7" title="PARTFUN3:func.7">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c46">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c47">p2</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c49">k</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun3.html#K9" title="PARTFUN3:func.9">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p2">(<span class="default"><a href="partfun3.html#K11" title="PARTFUN3:func.11">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K7" title="PARTFUN3:func.7">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c46">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c47">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14" title="PARTFUN3:func.14">/</a> <font color="Maroon" title="c46">m</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c50">x3</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c40">fx2</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun3.html#K9" title="PARTFUN3:func.9">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p4">(<span class="default"><a href="partfun3.html#K11" title="PARTFUN3:func.11">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K7" title="PARTFUN3:func.7">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c46">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c47">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14" title="PARTFUN3:func.14">/</a> <font color="Maroon" title="c46">m</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c51">y3</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c39">fy2</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun3.html#K9" title="PARTFUN3:func.9">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p4">(<span class="default"><a href="partfun3.html#K11" title="PARTFUN3:func.11">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K7" title="PARTFUN3:func.7">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c46">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c47">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14" title="PARTFUN3:func.14">/</a> <font color="Maroon" title="c46">m</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c52">X3</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c40">fx2</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun3.html#K9" title="PARTFUN3:func.9">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p4">(<span class="default"><a href="partfun3.html#K11" title="PARTFUN3:func.11">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K7" title="PARTFUN3:func.7">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c46">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c47">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14" title="PARTFUN3:func.14">/</a> <font color="Maroon" title="c46">m</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span>, <font color="Maroon" title="c53">Y3</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c39">fy2</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun3.html#K9" title="PARTFUN3:func.9">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p4">(<span class="default"><a href="partfun3.html#K11" title="PARTFUN3:func.11">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K7" title="PARTFUN3:func.7">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c46">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c47">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14" title="PARTFUN3:func.14">/</a> <font color="Maroon" title="c46">m</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c15">OK</font></span>)</span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c54">F</font> = <span class="p1"><a href="funct_3.html#K14" title="FUNCT_3:func.14">&lt;:</a><span class="default"><font color="Maroon" title="c52">X3</font>,<font color="Maroon" title="c53">Y3</font></span><a href="funct_3.html#K14" title="FUNCT_3:func.14">:&gt;</a></span>;<br/>
<a NAME="E62:22"/><span class="lab"><font color="Green" title="E37">A173</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span> holds  <span class="p1">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c18">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2"><a href="funct_3.html#K14" title="FUNCT_3:func.14">&lt;:</a><span class="default"><font color="Maroon" title="c52">X3</font>,<font color="Maroon" title="c53">Y3</font></span><a href="funct_3.html#K14" title="FUNCT_3:func.14">:&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Olive" title="b1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_18"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:22_18"/><span class="lab"><font color="Green" title="E38">A174</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K7" title="PARTFUN3:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c46">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c47">p2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c15">OK</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c55">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c18">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c55">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2"><a href="funct_3.html#K14" title="FUNCT_3:func.14">&lt;:</a><span class="default"><font color="Maroon" title="c52">X3</font>,<font color="Maroon" title="c53">Y3</font></span><a href="funct_3.html#K14" title="FUNCT_3:func.14">:&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c55">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c55">x</font></span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span><br/>

<a NAME="E2:22_18"/><span class="lab"><font color="Green" title="E39">A175</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c52">X3</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 <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c53">Y3</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 <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E3:22_18"/><span class="lab"><font color="Green" title="E40">A176</font></span>: 
not <font color="Maroon" title="c55">x</font> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a> <font color="Maroon" title="c18">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:22"><span class="lab"><font color="Green" title="E26">A16</font></span></a>, <a class="ref" href="abian.html#D5" title="ABIAN:def.5">ABIAN:def 5</a></span>;<br/>
<a NAME="E4:22_18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A177</font></span>: 
<font color="Maroon" title="c55">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c55">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#D4" title="ABIAN:def.4">ABIAN:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c56">y</font>, <font color="Maroon" title="c57">z</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E6:22_18"/><span class="lab"><font color="Green" title="E42">A178</font></span>: 
( <font color="Maroon" title="c56">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c55">x</font> &amp; <font color="Maroon" title="c57">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c55">x</font> )
 <span class="kw">and </span><br/><a NAME="E7:22_18"/><span class="lab"><font color="Green" title="E43">A179</font></span>: 
 <a href="brouwer.html#K4" title="BROUWER:func.4">HC</a> <font color="Maroon" title="c55">x</font>,<font color="Maroon" title="c18">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="brouwer.html#K3" title="BROUWER:func.3">HC</a> <font color="Maroon" title="c57">z</font>,<font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_18"><span class="lab"><font color="Green" title="E40">A176</font></span></a>, <a class="ref" href="brouwer.html#D4" target="_self" title="BROUWER:def.4">Def4</a></span>;<br/>
<a NAME="E8:22_18"/><span class="lab"><font color="Green" title="E44">A180</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:22"><span class="lab"><font color="Green" title="E35">A140</font></span></a>, <a class="txt" href="#E6:22_18"><span class="lab"><font color="Green" title="E42">A178</font></span></a>, <a class="txt" href="#E4:22_18"><span class="lab"><font color="Green" title="E41">A177</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/>
<a NAME="E9:22_18"/><span class="lab"><font color="Green" title="E45">A181</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">xo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">yo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:22"><span class="lab"><font color="Green" title="E35">A140</font></span></a>, <a class="txt" href="#E6:22_18"><span class="lab"><font color="Green" title="E42">A178</font></span></a>, <a class="txt" href="#E4:22_18"><span class="lab"><font color="Green" title="E41">A177</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c58">r1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c59">r2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c60">r3</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c61">r4</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c62">x9</font>, <font color="Maroon" title="c63">x10</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E11:22_18"/><span class="lab"><font color="Green" title="E46">A182</font></span>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c62">x9</font>,<font color="Maroon" title="c63">x10</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">and </span><br/><a NAME="E12:22_18"/><span class="lab"><font color="Green" title="E47">A183</font></span>: 
<font color="Maroon" title="c29">xo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c63">x10</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22"><span class="lab"><font color="Green" title="E16">A3</font></span></a></span>;<br/>
<a NAME="E13:22_18"/><span class="lab"><font color="Green" title="E48">A184</font></span>: 
<font color="Maroon" title="c57">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c63">x10</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22_18"><span class="lab"><font color="Green" title="E46">A182</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c64">x11</font>, <font color="Maroon" title="c65">x12</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E15:22_18"/><span class="lab"><font color="Green" title="E49">A185</font></span>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c64">x11</font>,<font color="Maroon" title="c65">x12</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">and </span><br/><a NAME="E16:22_18"/><span class="lab"><font color="Green" title="E50">A186</font></span>: 
<font color="Maroon" title="c28">yo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c65">x12</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22"><span class="lab"><font color="Green" title="E20">A9</font></span></a></span>;<br/>
<a NAME="E17:22_18"/><span class="lab"><font color="Green" title="E51">A187</font></span>: 
<font color="Maroon" title="c57">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c65">x12</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:22_18"><span class="lab"><font color="Green" title="E49">A185</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/>
<a NAME="E18:22_18"/>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c55">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c55">x</font></span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="brouwer.html#K1" title="BROUWER:func.1">DiffElems</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22_18"><span class="lab"><font color="Green" title="E42">A178</font></span></a>, <a class="txt" href="#E4:22_18"><span class="lab"><font color="Green" title="E41">A177</font></span></a></span>;<br/>
<a NAME="E19:22_18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A188</font></span>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">OK</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22_18"><span class="lab"><font color="Green" title="E42">A178</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E53">A189</font></span>: <font color="Maroon" title="c47">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></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"><font color="Maroon" title="c10">f1</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:22"><span class="lab"><font color="Green" title="E33">A52</font></span></a>, <a class="txt" href="#E55:22"><span class="lab"><font color="Green" title="E34">A139</font></span></a>, <a class="ref" href="valued_1.html#T13" title="VALUED_1:th.13">VALUED_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_18"><span class="lab"><font color="Green" title="E44">A180</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:22"><span class="lab"><font color="Green" title="E33">A52</font></span></a>, <a class="txt" href="#E19:22_18"><span class="lab"><font color="Green" title="E52">A188</font></span></a>, <a class="ref" href="valued_1.html#T1" title="VALUED_1:th.1">VALUED_1:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:22_18"><span class="lab"><font color="Green" title="E47">A183</font></span></a>, <a class="txt" href="#E13:22_18"><span class="lab"><font color="Green" title="E48">A184</font></span></a>, <a class="txt" href="#E16:22_18"><span class="lab"><font color="Green" title="E50">A186</font></span></a>, <a class="txt" href="#E17:22_18"><span class="lab"><font color="Green" title="E51">A187</font></span></a>, <a class="txt" href="#E9:22_18"><span class="lab"><font color="Green" title="E45">A181</font></span></a>, <a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></span>
;<br/>
<a NAME="E21:22_18"/><span class="lab"><font color="Green" title="E54">A190</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">dx</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:22"><span class="lab"><font color="Green" title="E35">A140</font></span></a>, <a class="txt" href="#E6:22_18"><span class="lab"><font color="Green" title="E42">A178</font></span></a>, <a class="txt" href="#E4:22_18"><span class="lab"><font color="Green" title="E41">A177</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c66">x7</font>, <font color="Maroon" title="c67">x8</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E23:22_18"/><span class="lab"><font color="Green" title="E55">A191</font></span>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c66">x7</font>,<font color="Maroon" title="c67">x8</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">and </span><br/><a NAME="E24:22_18"/><span class="lab"><font color="Green" title="E56">A192</font></span>: 
<font color="Maroon" title="c39">fy2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c67">x8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:22"><span class="lab"><font color="Green" title="E30">A22</font></span></a></span>;<br/>
<a NAME="E25:22_18"/><span class="lab"><font color="Green" title="E57">A193</font></span>: 
<font color="Maroon" title="c57">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c67">x8</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:22_18"><span class="lab"><font color="Green" title="E55">A191</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c68">x1</font>, <font color="Maroon" title="c69">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E27:22_18"/><span class="lab"><font color="Green" title="E58">A194</font></span>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c68">x1</font>,<font color="Maroon" title="c69">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">and </span><br/><a NAME="E28:22_18"/><span class="lab"><font color="Green" title="E59">A195</font></span>: 
<font color="Maroon" title="c40">fx2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c69">x2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22"><span class="lab"><font color="Green" title="E18">A6</font></span></a></span>;<br/>
<a NAME="E29:22_18"/><span class="lab"><font color="Green" title="E60">A196</font></span>: 
<font color="Maroon" title="c57">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c69">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:22_18"><span class="lab"><font color="Green" title="E58">A194</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c70">x3</font>, <font color="Maroon" title="c71">x4</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E31:22_18"/><span class="lab"><font color="Green" title="E61">A197</font></span>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c70">x3</font>,<font color="Maroon" title="c71">x4</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">and </span><br/><a NAME="E32:22_18"/><span class="lab"><font color="Green" title="E62">A198</font></span>: 
<font color="Maroon" title="c35">dx</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c70">x3</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c71">x4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:22"><span class="lab"><font color="Green" title="E32">A25</font></span></a></span>;<br/>
<a NAME="E33:22_18"/><span class="lab"><font color="Green" title="E63">A199</font></span>: 
( <font color="Maroon" title="c56">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c70">x3</font> &amp; <font color="Maroon" title="c57">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c71">x4</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:22_18"><span class="lab"><font color="Green" title="E61">A197</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c72">l</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>;<br/>
<a NAME="E34:22_18"/><span class="lab"><font color="Green" title="E64">A200</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></span>;<br/>
<a NAME="E35:22_18"/><span class="lab"><font color="Green" title="E65">A201</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">dy</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:22"><span class="lab"><font color="Green" title="E35">A140</font></span></a>, <a class="txt" href="#E6:22_18"><span class="lab"><font color="Green" title="E42">A178</font></span></a>, <a class="txt" href="#E4:22_18"><span class="lab"><font color="Green" title="E41">A177</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c73">x5</font>, <font color="Maroon" title="c74">x6</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E37:22_18"/><span class="lab"><font color="Green" title="E66">A202</font></span>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c73">x5</font>,<font color="Maroon" title="c74">x6</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">and </span><br/><a NAME="E38:22_18"/><span class="lab"><font color="Green" title="E67">A203</font></span>: 
<font color="Maroon" title="c34">dy</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c73">x5</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c74">x6</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:22"><span class="lab"><font color="Green" title="E28">A19</font></span></a></span>;<br/>
<a NAME="E39:22_18"/><span class="lab"><font color="Green" title="E68">A204</font></span>: 
( <font color="Maroon" title="c56">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c73">x5</font> &amp; <font color="Maroon" title="c57">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c74">x6</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:22_18"><span class="lab"><font color="Green" title="E66">A202</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/>
<span class="lab"><font color="Green" title="E69">A205</font></span>: <font color="Maroon" title="c46">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:22"><span class="lab"><font color="Green" title="E33">A52</font></span></a>, <a class="txt" href="#E19:22_18"><span class="lab"><font color="Green" title="E52">A188</font></span></a>, <a class="ref" href="valued_1.html#T1" title="VALUED_1:th.1">VALUED_1:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:22_18"><span class="lab"><font color="Green" title="E62">A198</font></span></a>, <a class="txt" href="#E33:22_18"><span class="lab"><font color="Green" title="E63">A199</font></span></a>, <a class="txt" href="#E38:22_18"><span class="lab"><font color="Green" title="E67">A203</font></span></a>, <a class="txt" href="#E39:22_18"><span class="lab"><font color="Green" title="E68">A204</font></span></a>, <a class="txt" href="#E21:22_18"><span class="lab"><font color="Green" title="E54">A190</font></span></a>, <a class="txt" href="#E35:22_18"><span class="lab"><font color="Green" title="E65">A201</font></span></a>, <a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></span>
;<br/>
<a NAME="E41:22_18"/><span class="lab"><font color="Green" title="E70">A206</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:22"><span class="lab"><font color="Green" title="E33">A52</font></span></a>, <a class="txt" href="#E19:22_18"><span class="lab"><font color="Green" title="E52">A188</font></span></a>, <a class="ref" href="valued_1.html#T1" title="VALUED_1:th.1">VALUED_1:1</a></span>;<br/>
<a NAME="E42:22_18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A207</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:22_18"><span class="lab"><font color="Green" title="E62">A198</font></span></a>, <a class="txt" href="#E33:22_18"><span class="lab"><font color="Green" title="E63">A199</font></span></a>, <a class="txt" href="#E38:22_18"><span class="lab"><font color="Green" title="E67">A203</font></span></a>, <a class="txt" href="#E39:22_18"><span class="lab"><font color="Green" title="E68">A204</font></span></a>, <a class="txt" href="#E12:22_18"><span class="lab"><font color="Green" title="E47">A183</font></span></a>, <a class="txt" href="#E13:22_18"><span class="lab"><font color="Green" title="E48">A184</font></span></a>, <a class="txt" href="#E16:22_18"><span class="lab"><font color="Green" title="E50">A186</font></span></a>, <a class="txt" href="#E17:22_18"><span class="lab"><font color="Green" title="E51">A187</font></span></a>, <a class="txt" href="#E21:22_18"><span class="lab"><font color="Green" title="E54">A190</font></span></a>, <a class="txt" href="#E35:22_18"><span class="lab"><font color="Green" title="E65">A201</font></span></a>, <a class="txt" href="#E9:22_18"><span class="lab"><font color="Green" title="E45">A181</font></span></a>, <a class="txt" href="#E34:22_18"><span class="lab"><font color="Green" title="E64">A200</font></span></a>, <a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></span>;<br/>
<a NAME="E43:22_18"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="partfun3.html#K11" title="PARTFUN3:func.11">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K7" title="PARTFUN3:func.7">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c46">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c47">p2</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c15">OK</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E72">A208</font></span>: <span class="p1">(<span class="default"><a href="partfun3.html#K11" title="PARTFUN3:func.11">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K7" title="PARTFUN3:func.7">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c46">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c47">p2</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> = 
 <a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K7" title="PARTFUN3:func.7">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c46">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c47">p2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:22"><span class="lab"><font color="Green" title="E33">A52</font></span></a>, <a class="txt" href="#E19:22_18"><span class="lab"><font color="Green" title="E52">A188</font></span></a>, <a class="ref" href="partfun3.html#D5" title="PARTFUN3:def.5">PARTFUN3:def 5</a></span>
<br/>.= 
 <a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c46">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c47">p2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:22"><span class="lab"><font color="Green" title="E33">A52</font></span></a>, <a class="txt" href="#E19:22_18"><span class="lab"><font color="Green" title="E52">A188</font></span></a>, <a class="txt" href="#E1:22_18"><span class="lab"><font color="Green" title="E38">A174</font></span></a>, <a class="ref" href="valued_1.html#T13" title="VALUED_1:th.13">VALUED_1:13</a></span>
<br/>.= 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:22_18"><span class="lab"><font color="Green" title="E69">A205</font></span></a>, <a class="txt" href="#E42:22_18"><span class="lab"><font color="Green" title="E71">A207</font></span></a>, <a class="txt" href="#E20:22_18"><span class="lab"><font color="Green" title="E53">A189</font></span></a>, <a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></span>
;<br/>
<a NAME="E45:22_18"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun3.html#K9" title="PARTFUN3:func.9">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p3">(<span class="default"><a href="partfun3.html#K11" title="PARTFUN3:func.11">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K7" title="PARTFUN3:func.7">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c46">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c47">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14" title="PARTFUN3:func.14">/</a> <font color="Maroon" title="c46">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c15">OK</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E73">A209</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun3.html#K9" title="PARTFUN3:func.9">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p3">(<span class="default"><a href="partfun3.html#K11" title="PARTFUN3:func.11">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K7" title="PARTFUN3:func.7">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c46">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c47">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14" title="PARTFUN3:func.14">/</a> <font color="Maroon" title="c46">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun3.html#K9" title="PARTFUN3:func.9">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p3">(<span class="default"><a href="partfun3.html#K11" title="PARTFUN3:func.11">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K7" title="PARTFUN3:func.7">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c46">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c47">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c46">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:22"><span class="lab"><font color="Green" title="E33">A52</font></span></a>, <a class="txt" href="#E19:22_18"><span class="lab"><font color="Green" title="E52">A188</font></span></a>, <a class="ref" href="rfunct_1.html#D4" title="RFUNCT_1:def.4">RFUNCT_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun3.html#K9" title="PARTFUN3:func.9">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p3">(<span class="default"><a href="partfun3.html#K11" title="PARTFUN3:func.11">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K7" title="PARTFUN3:func.7">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c46">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c47">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c46">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun3.html#K9" title="PARTFUN3:func.9">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span> <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="partfun3.html#K11" title="PARTFUN3:func.11">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K7" title="PARTFUN3:func.7">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c46">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c47">p2</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:22"><span class="lab"><font color="Green" title="E33">A52</font></span></a>, <a class="txt" href="#E19:22_18"><span class="lab"><font color="Green" title="E52">A188</font></span></a>, <a class="txt" href="#E40:22_18"><span class="lab"><font color="Green" title="E69">A205</font></span></a>, <a class="ref" href="valued_1.html#T1" title="VALUED_1:th.1">VALUED_1:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:22_18"><span class="lab"><font color="Green" title="E62">A198</font></span></a>, <a class="txt" href="#E33:22_18"><span class="lab"><font color="Green" title="E63">A199</font></span></a>, <a class="txt" href="#E38:22_18"><span class="lab"><font color="Green" title="E67">A203</font></span></a>, <a class="txt" href="#E39:22_18"><span class="lab"><font color="Green" title="E68">A204</font></span></a>, <a class="txt" href="#E12:22_18"><span class="lab"><font color="Green" title="E47">A183</font></span></a>, <a class="txt" href="#E13:22_18"><span class="lab"><font color="Green" title="E48">A184</font></span></a>, <a class="txt" href="#E16:22_18"><span class="lab"><font color="Green" title="E50">A186</font></span></a>, <a class="txt" href="#E17:22_18"><span class="lab"><font color="Green" title="E51">A187</font></span></a>, <a class="txt" href="#E21:22_18"><span class="lab"><font color="Green" title="E54">A190</font></span></a>, <a class="txt" href="#E35:22_18"><span class="lab"><font color="Green" title="E65">A201</font></span></a>, <a class="txt" href="#E9:22_18"><span class="lab"><font color="Green" title="E45">A181</font></span></a>, <a class="txt" href="#E34:22_18"><span class="lab"><font color="Green" title="E64">A200</font></span></a>, <a class="txt" href="#E41:22_18"><span class="lab"><font color="Green" title="E70">A206</font></span></a>, <a class="txt" href="#E44:22_18"><span class="lab"><font color="Green" title="E72">A208</font></span></a>, <a class="ref" href="valued_1.html#T8" title="VALUED_1:th.8">VALUED_1:8</a></span>
;<br/>
<span class="lab"><font color="Green" title="E74">A210</font></span>: <font color="Maroon" title="c53">Y3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c39">fy2</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="partfun3.html#K9" title="PARTFUN3:func.9">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p5">(<span class="default"><a href="partfun3.html#K11" title="PARTFUN3:func.11">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K7" title="PARTFUN3:func.7">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c46">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c47">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14" title="PARTFUN3:func.14">/</a> <font color="Maroon" title="c46">m</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:22"><span class="lab"><font color="Green" title="E33">A52</font></span></a>, <a class="txt" href="#E19:22_18"><span class="lab"><font color="Green" title="E52">A188</font></span></a>, <a class="ref" href="valued_1.html#T1" title="VALUED_1:th.1">VALUED_1:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="partfun3.html#K9" title="PARTFUN3:func.9">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p5">(<span class="default"><a href="partfun3.html#K11" title="PARTFUN3:func.11">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K7" title="PARTFUN3:func.7">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c46">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c47">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14" title="PARTFUN3:func.14">/</a> <font color="Maroon" title="c46">m</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:22"><span class="lab"><font color="Green" title="E35">A140</font></span></a>, <a class="txt" href="#E6:22_18"><span class="lab"><font color="Green" title="E42">A178</font></span></a>, <a class="txt" href="#E4:22_18"><span class="lab"><font color="Green" title="E41">A177</font></span></a>, <a class="txt" href="#E24:22_18"><span class="lab"><font color="Green" title="E56">A192</font></span></a>, <a class="txt" href="#E25:22_18"><span class="lab"><font color="Green" title="E57">A193</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:22_18"><span class="lab"><font color="Green" title="E67">A203</font></span></a>, <a class="txt" href="#E39:22_18"><span class="lab"><font color="Green" title="E68">A204</font></span></a>, <a class="txt" href="#E35:22_18"><span class="lab"><font color="Green" title="E65">A201</font></span></a>, <a class="txt" href="#E46:22_18"><span class="lab"><font color="Green" title="E73">A209</font></span></a>, <a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></span>
;<br/>
<span class="lab"><font color="Green" title="E75">A211</font></span>: <font color="Maroon" title="c52">X3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c40">fx2</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="partfun3.html#K9" title="PARTFUN3:func.9">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p5">(<span class="default"><a href="partfun3.html#K11" title="PARTFUN3:func.11">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K7" title="PARTFUN3:func.7">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c46">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c47">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14" title="PARTFUN3:func.14">/</a> <font color="Maroon" title="c46">m</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:22"><span class="lab"><font color="Green" title="E33">A52</font></span></a>, <a class="txt" href="#E19:22_18"><span class="lab"><font color="Green" title="E52">A188</font></span></a>, <a class="ref" href="valued_1.html#T1" title="VALUED_1:th.1">VALUED_1:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="partfun3.html#K9" title="PARTFUN3:func.9">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p5">(<span class="default"><a href="partfun3.html#K11" title="PARTFUN3:func.11">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c29">xo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c28">yo</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c34">dy</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K7" title="PARTFUN3:func.7">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c46">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c47">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14" title="PARTFUN3:func.14">/</a> <font color="Maroon" title="c46">m</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c35">dx</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:22"><span class="lab"><font color="Green" title="E35">A140</font></span></a>, <a class="txt" href="#E6:22_18"><span class="lab"><font color="Green" title="E42">A178</font></span></a>, <a class="txt" href="#E4:22_18"><span class="lab"><font color="Green" title="E41">A177</font></span></a>, <a class="txt" href="#E28:22_18"><span class="lab"><font color="Green" title="E59">A195</font></span></a>, <a class="txt" href="#E29:22_18"><span class="lab"><font color="Green" title="E60">A196</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:22_18"><span class="lab"><font color="Green" title="E62">A198</font></span></a>, <a class="txt" href="#E33:22_18"><span class="lab"><font color="Green" title="E63">A199</font></span></a>, <a class="txt" href="#E21:22_18"><span class="lab"><font color="Green" title="E54">A190</font></span></a>, <a class="txt" href="#E46:22_18"><span class="lab"><font color="Green" title="E73">A209</font></span></a>, <a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></span>
;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c18">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c55">x</font> = 
 <a href="brouwer.html#K4" title="BROUWER:func.4">HC</a> <font color="Maroon" title="c55">x</font>,<font color="Maroon" title="c18">f</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="brouwer.html#D5" target="_self" title="BROUWER:def.5">Def5</a></span>
<br/>.= 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22_18"><span class="lab"><font color="Green" title="E42">A178</font></span></a>, <a class="txt" href="#E7:22_18"><span class="lab"><font color="Green" title="E43">A179</font></span></a>, <a class="txt" href="#E4:22_18"><span class="lab"><font color="Green" title="E41">A177</font></span></a>, <a class="ref" href="brouwer.html#T8" target="_self" title="BROUWER:th.8">Th8</a></span>
<br/>.= 
<a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c52">X3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c53">Y3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:22_18"><span class="lab"><font color="Green" title="E75">A211</font></span></a>, <a class="txt" href="#E47:22_18"><span class="lab"><font color="Green" title="E74">A210</font></span></a>, <a class="ref" href="topreala.html#D2" title="TOPREALA:def.2">TOPREALA:def 2</a></span>
<br/>.= 
<a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K14" title="FUNCT_3:func.14">&lt;:</a><span class="default"><font color="Maroon" title="c52">X3</font>,<font color="Maroon" title="c53">Y3</font></span><a href="funct_3.html#K14" title="FUNCT_3:func.14">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:22"><span class="lab"><font color="Green" title="E33">A52</font></span></a>, <a class="txt" href="#E19:22_18"><span class="lab"><font color="Green" title="E52">A188</font></span></a>, <a class="txt" href="#E2:22_18"><span class="lab"><font color="Green" title="E39">A175</font></span></a>, <a class="ref" href="funct_3.html#T69" title="FUNCT_3:th.69">FUNCT_3:69</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2"><a href="funct_3.html#K14" title="FUNCT_3:func.14">&lt;:</a><span class="default"><font color="Maroon" title="c52">X3</font>,<font color="Maroon" title="c53">Y3</font></span><a href="funct_3.html#K14" title="FUNCT_3:func.14">:&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c55">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c55">x</font></span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:22"><span class="lab"><font color="Green" title="E33">A52</font></span></a>, <a class="txt" href="#E6:22_18"><span class="lab"><font color="Green" title="E42">A178</font></span></a>, <a class="txt" href="#E19:22_18"><span class="lab"><font color="Green" title="E52">A188</font></span></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E63:22"/>
( <font color="Maroon" title="c52">X3</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; <font color="Maroon" title="c53">Y3</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal6.html#T83" title="TOPREAL6:th.83">TOPREAL6:83</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c55">F</font> = <span class="p1"><a href="funct_3.html#K14" title="FUNCT_3:func.14">&lt;:</a><span class="default"><font color="Maroon" title="c52">X3</font>,<font color="Maroon" title="c53">Y3</font></span><a href="funct_3.html#K14" title="FUNCT_3:func.14">:&gt;</a></span> as   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c15">OK</font></span>)</span>,<span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> ,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> </span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow12.html#T41" title="YELLOW12:th.41">YELLOW12:41</a></span>;<br/>
<a NAME="E65:22"/>
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span><br/> for <font color="Olive" title="b2">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span>  st <span class="p1">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c18">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">V</font> &amp; <font color="Olive" title="b2">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/> ex <font color="Olive" title="b3">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">W</font> &amp; <font color="Olive" title="b3">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <span class="p1">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c18">f</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b3">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_19"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c56">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span>  st <span class="p1">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c18">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c56">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b1">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/> ex <font color="Olive" title="b2">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span> st <br/>( <font color="Maroon" title="c56">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">W</font> &amp; <font color="Olive" title="b2">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <span class="p1">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c18">f</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">V</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c57">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c18">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c56">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c57">V</font> &amp; <font color="Maroon" title="c57">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies  ex <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span> st <br/>( <font color="Maroon" title="c56">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <span class="p1">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c18">f</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c57">V</font> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:22_19"/><span class="lab"><font color="Green" title="E38">A212</font></span>: 
<span class="p1">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c18">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c56">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c57">V</font>
 <span class="kw">and </span><br/><a NAME="E2:22_19"/><span class="lab"><font color="Green" title="E39">A213</font></span>: 
<font color="Maroon" title="c57">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span> st <br/>( <font color="Maroon" title="c56">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <span class="p1">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c18">f</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c57">V</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c58">V1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E4:22_19"/><span class="lab"><font color="Green" title="E40">A214</font></span>: 
<font color="Maroon" title="c58">V1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E5:22_19"/><span class="lab"><font color="Green" title="E41">A215</font></span>: 
<font color="Maroon" title="c58">V1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c57">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_19"><span class="lab"><font color="Green" title="E39">A213</font></span></a>, <a class="ref" href="tops_2.html#T32" title="TOPS_2:th.32">TOPS_2:32</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c59">p1</font> = <font color="Maroon" title="c56">p</font>, <font color="Maroon" title="c60">fp</font> = <font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c56">p</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T55" title="PRE_TOPC:th.55">PRE_TOPC:55</a></span>;<br/>
<a NAME="E7:22_19"/><span class="lab"><font color="Green" title="E42">A216</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreala.html#T56" title="TOPREALA:th.56">TOPREALA:56</a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/>
<a NAME="E8:22_19"/>
<a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreala.html#T56" title="TOPREALA:th.56">TOPREALA:56</a>, <a class="ref" href="tops_2.html#T70" title="TOPS_2:th.70">TOPS_2:70</a></span>;<br/>
<a NAME="E9:22_19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A217</font></span>: 
<span class="p1">(<span class="default"><a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c58">V1</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="#E4:22_19"><span class="lab"><font color="Green" title="E40">A214</font></span></a>, <a class="ref" href="topgrp_1.html#T25" title="TOPGRP_1:th.25">TOPGRP_1:25</a></span>;<br/>
<a NAME="E10:22_19"/>
not <font color="Maroon" title="c56">p</font> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a> <font color="Maroon" title="c18">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:22"><span class="lab"><font color="Green" title="E26">A16</font></span></a>, <a class="ref" href="abian.html#D5" title="ABIAN:def.5">ABIAN:def 5</a></span>;<br/>
<a NAME="E11:22_19"/><span class="kw">then </span>
<font color="Maroon" title="c56">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c56">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#D4" title="ABIAN:def.4">ABIAN:def 4</a></span>;<br/>
<a NAME="E12:22_19"/><span class="kw">then </span>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c59">p1</font>,<font color="Maroon" title="c60">fp</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="brouwer.html#K1" title="BROUWER:func.1">DiffElems</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
;<br/>
<a NAME="E13:22_19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A218</font></span>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c56">p</font></span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">OK</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E14:22_19"/>
<span class="p1">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c18">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c56">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c56">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c56">p</font></span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E62:22"><span class="lab"><font color="Green" title="E37">A173</font></span></a></span>;<br/>
<a NAME="E15:22_19"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c59">p1</font>,<font color="Maroon" title="c60">fp</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c58">V1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_19"><span class="lab"><font color="Green" title="E38">A212</font></span></a>, <a class="txt" href="#E5:22_19"><span class="lab"><font color="Green" title="E41">A215</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E16:22_19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A219</font></span>: 
<span class="p1">(<span class="default"><a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c59">p1</font>,<font color="Maroon" title="c60">fp</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c58">V1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></span>;<br/>
<a NAME="E17:22_19"/><span class="lab"><font color="Green" title="E46">A220</font></span>: 
 <a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreala.html#T56" title="TOPREALA:th.56">TOPREALA:56</a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/>
<a NAME="E18:22_19"/><span class="lab"><font color="Green" title="E47">A221</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> ,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> </span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E19:22_19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A222</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c55">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a> 
 
;<br/>
<a NAME="E20:22_19"/><span class="kw">then </span>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c55">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 <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c15">OK</font></span>)</span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c55">F</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></span>;<br/>
<a NAME="E21:22_19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A223</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c55">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c59">p1</font>,<font color="Maroon" title="c60">fp</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c58">V1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:22"><span class="lab"><font color="Green" title="E33">A52</font></span></a>, <a class="txt" href="#E13:22_19"><span class="lab"><font color="Green" title="E44">A218</font></span></a>, <a class="txt" href="#E16:22_19"><span class="lab"><font color="Green" title="E45">A219</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>;<br/>
<a NAME="E22:22_19"/><span class="lab"><font color="Green" title="E50">A224</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c55">F</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p3">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c55">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_19_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c61">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c61">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c55">F</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p3">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c61">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c55">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c61">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_19_1"/><span class="lab"><font color="Green" title="E51">A225</font></span>: 
<font color="Maroon" title="c61">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c55">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p3">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c61">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c55">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c61">x</font></span><br/>

<a NAME="E2:22_19_1"/><span class="lab"><font color="Green" title="E52">A226</font></span>: 
<font color="Maroon" title="c55">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c61">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c55">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_19_1"><span class="lab"><font color="Green" title="E51">A225</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p3">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c61">x</font> = 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c55">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c61">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_19_1"><span class="lab"><font color="Green" title="E51">A225</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
<br/>.= 
<font color="Maroon" title="c55">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c61">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:22_19"><span class="lab"><font color="Green" title="E47">A221</font></span></a>, <a class="txt" href="#E2:22_19_1"><span class="lab"><font color="Green" title="E52">A226</font></span></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E23:22_19"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></span>;<br/>
<a NAME="E24:22_19"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p3">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c55">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:22_19"><span class="lab"><font color="Green" title="E48">A222</font></span></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></span>;<br/>
<a NAME="E25:22_19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A227</font></span>: 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c55">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c55">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:22_19"><span class="lab"><font color="Green" title="E50">A224</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></span>;<br/>
<span class="p1">(<span class="default"><a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c55">F</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c55">F</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c55">F</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_19"><span class="lab"><font color="Green" title="E42">A216</font></span></a>, <a class="txt" href="#E17:22_19"><span class="lab"><font color="Green" title="E46">A220</font></span></a>, <a class="ref" href="tops_2.html#T65" title="TOPS_2:th.65">TOPS_2:65</a></span>
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c61">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c15">OK</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E28:22_19"/><span class="lab"><font color="Green" title="E52">A228</font></span>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c59">p1</font>,<font color="Maroon" title="c60">fp</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c61">W</font>
 <span class="kw">and </span><br/><a NAME="E29:22_19"/><span class="lab"><font color="Green" title="E53">A229</font></span>: 
<font color="Maroon" title="c61">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E30:22_19"/><span class="lab"><font color="Green" title="E54">A230</font></span>: 
<font color="Maroon" title="c55">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c61">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c58">V1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:22"><span class="lab"><font color="Green" title="E33">A52</font></span></a>, <a class="txt" href="#E13:22_19"><span class="lab"><font color="Green" title="E44">A218</font></span></a>, <a class="txt" href="#E9:22_19"><span class="lab"><font color="Green" title="E43">A217</font></span></a>, <a class="txt" href="#E25:22_19"><span class="lab"><font color="Green" title="E51">A227</font></span></a>, <a class="txt" href="#E21:22_19"><span class="lab"><font color="Green" title="E49">A223</font></span></a>, <a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c62">WW</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E32:22_19"/><span class="lab"><font color="Green" title="E55">A231</font></span>: 
<font color="Maroon" title="c62">WW</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E33:22_19"/><span class="lab"><font color="Green" title="E56">A232</font></span>: 
<font color="Maroon" title="c62">WW</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c61">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:22_19"><span class="lab"><font color="Green" title="E53">A229</font></span></a>, <a class="ref" href="tops_2.html#T32" title="TOPS_2:th.32">TOPS_2:32</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c63">SF</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E35:22_19"/><span class="lab"><font color="Green" title="E57">A233</font></span>: 
<font color="Maroon" title="c62">WW</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c63">SF</font>
 <span class="kw">and </span><br/><a NAME="E36:22_19"/><span class="lab"><font color="Green" title="E58">A234</font></span>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c63">SF</font> holds <br/> ex <font color="Olive" title="b2">X1</font>, <font color="Olive" title="b3">Y1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b2">X1</font>,<font color="Olive" title="b3">Y1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> &amp; <font color="Olive" title="b2">X1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive" title="b3">Y1</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="#E32:22_19"><span class="lab"><font color="Green" title="E55">A231</font></span></a>, <a class="ref" href="borsuk_1.html#T45" title="BORSUK_1:th.45">BORSUK_1:45</a></span>;<br/>
<a NAME="E37:22_19"/>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c59">p1</font>,<font color="Maroon" title="c60">fp</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c62">WW</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:22_19"><span class="lab"><font color="Green" title="E52">A228</font></span></a>, <a class="txt" href="#E33:22_19"><span class="lab"><font color="Green" title="E56">A232</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c64">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E39:22_19"/><span class="lab"><font color="Green" title="E59">A235</font></span>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c59">p1</font>,<font color="Maroon" title="c60">fp</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c64">Z</font>
 <span class="kw">and </span><br/><a NAME="E40:22_19"/><span class="lab"><font color="Green" title="E60">A236</font></span>: 
<font color="Maroon" title="c64">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c63">SF</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:22_19"><span class="lab"><font color="Green" title="E57">A233</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c65">ZZ</font> = <font color="Maroon" title="c64">Z</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span>;<br/>
<a NAME="E41:22_19"/>
<font color="Maroon" title="c64">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c62">WW</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:22_19"><span class="lab"><font color="Green" title="E57">A233</font></span></a>, <a class="txt" href="#E40:22_19"><span class="lab"><font color="Green" title="E60">A236</font></span></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></span>;<br/>
<a NAME="E42:22_19"/><span class="kw">then </span>
<font color="Maroon" title="c64">Z</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c62">WW</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></span>;<br/>
<a NAME="E43:22_19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A237</font></span>: 
<font color="Maroon" title="c55">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c64">Z</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><span class="p4"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p5">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c55">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c61">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:22_19"><span class="lab"><font color="Green" title="E56">A232</font></span></a>, <a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c66">X1</font>, <font color="Maroon" title="c67">Y1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E45:22_19"/><span class="lab"><font color="Green" title="E62">A238</font></span>: 
<font color="Maroon" title="c64">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c66">X1</font>,<font color="Maroon" title="c67">Y1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E46:22_19"/><span class="lab"><font color="Green" title="E63">A239</font></span>: 
( <font color="Maroon" title="c66">X1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c67">Y1</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="#E36:22_19"><span class="lab"><font color="Green" title="E58">A234</font></span></a>, <a class="txt" href="#E40:22_19"><span class="lab"><font color="Green" title="E60">A236</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c68">XX</font> = <font color="Maroon" title="c66">X1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span></span>)</span>, <font color="Maroon" title="c69">YY</font> = <font color="Maroon" title="c67">Y1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span></span>)</span> as   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:22_19"><span class="lab"><font color="Green" title="E63">A239</font></span></a>, <a class="ref" href="tops_2.html#T32" title="TOPS_2:th.32">TOPS_2:32</a></span>;<br/>
<a NAME="E48:22_19"/>
<font color="Maroon" title="c60">fp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c67">Y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:22_19"><span class="lab"><font color="Green" title="E59">A235</font></span></a>, <a class="txt" href="#E45:22_19"><span class="lab"><font color="Green" title="E62">A238</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E49:22_19"/><span class="kw">then </span>
<font color="Maroon" title="c60">fp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c69">YY</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c70">M</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E51:22_19"/><span class="lab"><font color="Green" title="E64">A240</font></span>: 
<font color="Maroon" title="c56">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c70">M</font>
 <span class="kw">and </span><br/><a NAME="E52:22_19"/><span class="lab"><font color="Green" title="E65">A241</font></span>: 
<font color="Maroon" title="c70">M</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E53:22_19"/><span class="lab"><font color="Green" title="E66">A242</font></span>: 
<font color="Maroon" title="c18">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c70">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c69">YY</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c71">W1</font> = <font color="Maroon" title="c68">XX</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c70">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c56">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c71">W1</font> &amp; <font color="Maroon" title="c71">W1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <span class="p1">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c18">f</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c71">W1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c57">V</font> )</span><br/>

<a NAME="E54:22_19"/>
<font color="Maroon" title="c56">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c66">X1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:22_19"><span class="lab"><font color="Green" title="E59">A235</font></span></a>, <a class="txt" href="#E45:22_19"><span class="lab"><font color="Green" title="E62">A238</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E55:22_19"/><span class="kw">then </span>
<font color="Maroon" title="c56">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c68">XX</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E56:22_19"/>
<font color="Maroon" title="c56">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c71">W1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:22_19"><span class="lab"><font color="Green" title="E64">A240</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c71">W1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <span class="p1">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c18">f</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c71">W1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c57">V</font> )</span><br/>

<span class="kw">thus </span><a NAME="E57:22_19"/>
<font color="Maroon" title="c71">W1</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="#E52:22_19"><span class="lab"><font color="Green" title="E65">A241</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c18">f</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c71">W1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c57">V</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c72">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c72">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c18">f</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c71">W1</font> or <font color="Maroon" title="c72">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c57">V</font> )</span><br/>

<span class="kw">assume </span><a NAME="E58:22_19"/>
<font color="Maroon" title="c72">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c18">f</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c71">W1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c72">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c57">V</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c73">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">r</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E60:22_19"/><span class="lab"><font color="Green" title="E67">A243</font></span>: 
<font color="Maroon" title="c73">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c71">W1</font>
 <span class="kw">and </span><br/><a NAME="E61:22_19"/><span class="lab"><font color="Green" title="E68">A244</font></span>: 
<font color="Maroon" title="c72">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c18">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c73">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c74">a1</font> = <font color="Maroon" title="c73">a</font>, <font color="Maroon" title="c75">fa</font> = <font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c73">a</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T55" title="PRE_TOPC:th.55">PRE_TOPC:55</a></span>;<br/>
<a NAME="E63:22_19"/>
<font color="Maroon" title="c73">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c70">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:22_19"><span class="lab"><font color="Green" title="E67">A243</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E64:22_19"/><span class="kw">then </span>
<font color="Maroon" title="c75">fa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c70">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></span>;<br/>
<a NAME="E65:22_19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A245</font></span>: 
<font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c73">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c67">Y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:22_19"><span class="lab"><font color="Green" title="E66">A242</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E66:22_19"/>
not <font color="Maroon" title="c73">a</font> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a> <font color="Maroon" title="c18">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:22"><span class="lab"><font color="Green" title="E26">A16</font></span></a>, <a class="ref" href="abian.html#D5" title="ABIAN:def.5">ABIAN:def 5</a></span>;<br/>
<a NAME="E67:22_19"/><span class="kw">then </span>
<font color="Maroon" title="c73">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c73">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#D4" title="ABIAN:def.4">ABIAN:def 4</a></span>;<br/>
<a NAME="E68:22_19"/><span class="kw">then </span>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c74">a1</font>,<font color="Maroon" title="c75">fa</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="brouwer.html#K1" title="BROUWER:func.1">DiffElems</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
;<br/>
<a NAME="E69:22_19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A246</font></span>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c73">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c73">a</font></span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">OK</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E70:22_19"/>
<font color="Maroon" title="c73">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c68">XX</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:22_19"><span class="lab"><font color="Green" title="E67">A243</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E71:22_19"/><span class="kw">then </span>
<font color="Maroon" title="c73">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c66">X1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E72:22_19"/><span class="kw">then </span>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c73">a</font>,<font color="Maroon" title="c75">fa</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c64">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:22_19"><span class="lab"><font color="Green" title="E62">A238</font></span></a>, <a class="txt" href="#E65:22_19"><span class="lab"><font color="Green" title="E69">A245</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E73:22_19"/><span class="kw">then </span>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c73">a</font>,<font color="Maroon" title="c75">fa</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c64">Z</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:22"><span class="lab"><font color="Green" title="E33">A52</font></span></a>, <a class="txt" href="#E69:22_19"><span class="lab"><font color="Green" title="E70">A246</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E74:22_19"/><span class="kw">then </span>
<font color="Maroon" title="c55">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c74">a1</font>,<font color="Maroon" title="c75">fa</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c55">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c64">Z</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><span class="p4"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p5">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c15">OK</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></span>;<br/>
<a NAME="E75:22_19"/><span class="kw">then </span>
<font color="Maroon" title="c55">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c74">a1</font>,<font color="Maroon" title="c75">fa</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c55">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c61">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:22_19"><span class="lab"><font color="Green" title="E61">A237</font></span></a></span>;<br/>
<a NAME="E76:22_19"/><span class="kw">then </span>
<a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c55">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c74">a1</font>,<font color="Maroon" title="c75">fa</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c58">V1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:22_19"><span class="lab"><font color="Green" title="E54">A230</font></span></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></span>;<br/>
<a NAME="E77:22_19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A247</font></span>: 
<span class="p1">(<span class="default"><a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c74">a1</font>,<font color="Maroon" title="c75">fa</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c58">V1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:22"><span class="lab"><font color="Green" title="E33">A52</font></span></a>, <a class="txt" href="#E69:22_19"><span class="lab"><font color="Green" title="E70">A246</font></span></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>;<br/>
<a NAME="E78:22_19"/>
( <a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_19"><span class="lab"><font color="Green" title="E42">A216</font></span></a>, <a class="txt" href="#E17:22_19"><span class="lab"><font color="Green" title="E46">A220</font></span></a>, <a class="ref" href="tops_2.html#T62" title="TOPS_2:th.62">TOPS_2:62</a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></span>;<br/>
<a NAME="E79:22_19"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c74">a1</font>,<font color="Maroon" title="c75">fa</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c58">V1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:22_19"><span class="lab"><font color="Green" title="E46">A220</font></span></a>, <a class="txt" href="#E77:22_19"><span class="lab"><font color="Green" title="E71">A247</font></span></a>, <a class="ref" href="partfun3.html#T1" title="PARTFUN3:th.1">PARTFUN3:1</a></span>;<br/>
<a NAME="E80:22_19"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c18">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c73">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c58">V1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E62:22"><span class="lab"><font color="Green" title="E37">A173</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E81:22_19"/>
<font color="Maroon" title="c72">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c57">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_19"><span class="lab"><font color="Green" title="E41">A215</font></span></a>, <a class="txt" href="#E61:22_19"><span class="lab"><font color="Green" title="E68">A244</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E66:22"/>
 <a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c18">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
