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

<b>let </b><font color="Maroon" title="c2">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:()"><i><font color="Red">::  thesis: </font></i></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="c2">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="c2">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/><b>let </b><font color="Maroon" title="c3">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:()"><i><font color="Red">::  thesis: </font></i></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="c3">o</font>,<font color="Maroon" title="c2">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="c3">o</font>,<font color="Maroon" title="c2">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/>



<b>set </b><font color="Maroon" title="c4">D2</font> =  <a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c2">r</font>;<br/>
<b>set </b><font color="Maroon" title="c5">S1</font> =  <a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c2">r</font>;<br/>
<b>set </b><font color="Maroon" title="c6">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#K4" title="BORSUK_1:func.4">[:</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="c3">o</font>,<font color="Maroon" title="c2">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="c3">o</font>,<font color="Maroon" title="c2">r</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>;<br/>
<b>consider </b><font color="Maroon" title="c7">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="c3">o</font>,<font color="Maroon" title="c2">r</font></span>)</span>;<br/>
<a NAME="E2:22"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c7">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="c3">o</font>,<font color="Maroon" title="c2">r</font></span>)</span>
 
;<br/>
<a NAME="E3:22"/><i><font color="Green" title="E16">A2</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c2">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="c3">o</font>,<font color="Maroon" title="c2">r</font>
 
<b>by </b><i><a class="ref" href="toprealb.html#T9" title="TOPREALB:th.9">TOPREALB:9</a></i>;<br/>
<a NAME="E4:22"/><i><font color="Green" title="E17">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c2">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="c3">o</font>,<font color="Maroon" title="c2">r</font>
 
<b>by </b><i><a class="ref" href="brouwer.html#T3" target="_self" title="BROUWER:th.3">Th3</a></i>;<br/>
<a NAME="E5:22"/>
 <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c2">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="c3">o</font>,<font color="Maroon" title="c2">r</font>
 
<b>by </b><i><a class="ref" href="topreal9.html#T17" title="TOPREAL9:th.17">TOPREAL9:17</a></i>;<br/>
<a NAME="E6:22"/><b>then </b><i><font color="Green" title="E18">A4</font></i>: 
<font color="Maroon" title="c7">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="c3">o</font>,<font color="Maroon" title="c2">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:22"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="#E3:22"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="brouwer.html#T3" target="_self" title="BROUWER:th.3">Th3</a></i>;<br/>
<a NAME="E7:22"/><i><font color="Green" title="E19">A5</font></i>: 
<font color="Maroon" title="c7">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>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T55" title="PRE_TOPC:th.55">PRE_TOPC:55</a></i>;<br/>
<i><font color="Green" title="E20">A6</font></i>: <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="c3">o</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c3">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="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> 2</span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T46" title="EUCLID:th.46">EUCLID:46</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="toprns_1.html#T24" title="TOPRNS_1:th.24">TOPRNS_1:24</a></i>
;<br/>
<a NAME="E9:22"/><b>then </b>
<font color="Maroon" title="c3">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="c3">o</font>,<font color="Maroon" title="c2">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:22"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="topreal9.html#T8" title="TOPREAL9:th.8">TOPREAL9:8</a></i>;<br/>
<a NAME="E10:22"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c7">s</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c3">o</font>,<font color="Maroon" title="c2">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="c3">o</font>,<font color="Maroon" title="c2">r</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E6:22"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E11:22"/><b>then </b><i><font color="Green" title="E21">A7</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c7">s</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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#K4" title="BORSUK_1:func.4">[:</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="c3">o</font>,<font color="Maroon" title="c2">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="c3">o</font>,<font color="Maroon" title="c2">r</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:22_2"/><i><font color="Green" title="E22">A8</font></i>: 
<font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:22_2"/><i><font color="Green" title="E23">A9</font></i>: 
 <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c2">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="c3">o</font>,<font color="Maroon" title="c2">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T19" title="TOPREAL9:th.19">TOPREAL9:19</a></i>;<br/><a NAME="E3:22_2"/>
<font color="Maroon" title="c3">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="c3">o</font>,<font color="Maroon" title="c2">r</font>
 <b>by </b><i><a class="txt" href="#E8:22"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="topreal9.html#T7" title="TOPREAL9:th.7">TOPREAL9:7</a></i>;<br/><b>hence </b><a NAME="E4:22_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:22"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="#E1:22_2"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="#E2:22_2"><i><font color="Green" title="E23">A9</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E13:22"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c7">s</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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>
 
<b>by </b><i><a class="txt" href="#E7:22"><i><font color="Green" title="E19">A5</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">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#K4" title="BORSUK_1:func.4">[:</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="c3">o</font>,<font color="Maroon" title="c2">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="c3">o</font>,<font color="Maroon" title="c2">r</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> <b>by </b><i><a class="txt" href="#E11:22"><i><font color="Green" title="E21">A7</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">TD</font> = <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="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#K4" title="BORSUK_1:func.4">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">OK</font>;<br/>
<b>let </b><font color="Maroon" title="c10">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="c3">o</font>,<font color="Maroon" title="c2">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="c3">o</font>,<font color="Maroon" title="c2">r</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">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="c10">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )</span><br/>

<b>assume </b><a NAME="E15:22"/><i><font color="Green" title="E22">A10</font></i>: 
<font color="Maroon" title="c10">f</font> <a href="abian.html#NR4" title="ABIAN:NR.4">has_no_fixpoint</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c10">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a></span><br/>

<b>set </b><font color="Maroon" title="c11">g</font> =  <a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c10">f</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> );<br/>
<a NAME="E16:22"/><i><font color="Green" title="E23">A11</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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="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#K4" title="BORSUK_1:func.4">:]</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_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">r</font> 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="c12">x</font>,<font color="Olive" title="b1">r</font>]</span><br/>

<b>consider </b><font color="Maroon" title="c13">x1</font>, <font color="Maroon" title="c14">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><b> such that </b><br/><a NAME="E2:22_3"/><i><font color="Green" title="E24">A12</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c14">x2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50" title="BORSUK_1:th.50">BORSUK_1:50</a></i>;<br/>
<b>take </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">x2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">x2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>]</span><br/>

<b>take </b>
<font color="Maroon" title="c13">x1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">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="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c13">x1</font>,<font color="Olive" title="b1">x2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">x2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> )</span><br/>

<b>take </b>
<font color="Maroon" title="c14">x2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c14">x2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">x2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> )</span><br/>

<b>thus </b><a NAME="E3:22_3"/>
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c14">x2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">x2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="#E2:22_3"><i><font color="Green" title="E24">A12</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c12">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span><b> such that </b><br/><a NAME="E18:22"/><i><font color="Green" title="E24">A13</font></i>: 
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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c12">xo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E16:22"><i><font color="Green" title="E23">A11</font></i></a>);<br/></i>
<b>defpred </b><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> ] <b>means </b> 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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> );<br/>
<a NAME="E19:22"/><i><font color="Green" title="E25">A14</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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="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#K4" title="BORSUK_1:func.4">:]</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_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">r</font> 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="c13">x</font>,<font color="Olive" title="b1">r</font>]</span><br/>

<b>consider </b><font color="Maroon" title="c14">x1</font>, <font color="Maroon" title="c15">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><b> such that </b><br/><a NAME="E2:22_4"/><i><font color="Green" title="E26">A15</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c14">x1</font>,<font color="Maroon" title="c15">x2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50" title="BORSUK_1:th.50">BORSUK_1:50</a></i>;<br/>
<b>take </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">x2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">x2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>]</span><br/>

<b>take </b>
<font color="Maroon" title="c14">x1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">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="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c14">x1</font>,<font color="Olive" title="b1">x2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c15">x2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> )</span><br/>

<b>take </b>
<font color="Maroon" title="c15">x2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c14">x1</font>,<font color="Maroon" title="c15">x2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c15">x2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> )</span><br/>

<b>thus </b><a NAME="E3:22_4"/>
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c14">x1</font>,<font color="Maroon" title="c15">x2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c15">x2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="#E2:22_4"><i><font color="Green" title="E26">A15</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c13">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span><b> such that </b><br/><a NAME="E21:22"/><i><font color="Green" title="E26">A16</font></i>: 
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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c13">yo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E19:22"><i><font color="Green" title="E25">A14</font></i></a>);<br/></i>
<b>defpred </b><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> ] <b>means </b> 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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`1</a> </span>)</span> );<br/>
<a NAME="E22:22"/><i><font color="Green" title="E27">A17</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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="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#K4" title="BORSUK_1:func.4">:]</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_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">r</font> 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="c14">x</font>,<font color="Olive" title="b1">r</font>]</span><br/>

<b>consider </b><font color="Maroon" title="c15">x1</font>, <font color="Maroon" title="c16">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><b> such that </b><br/><a NAME="E2:22_5"/><i><font color="Green" title="E28">A18</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c15">x1</font>,<font color="Maroon" title="c16">x2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50" title="BORSUK_1:th.50">BORSUK_1:50</a></i>;<br/>
<b>take </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">x1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c16">x2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c14">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">x1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c16">x2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>]</span><br/>

<b>take </b>
<font color="Maroon" title="c15">x1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">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="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c15">x1</font>,<font color="Olive" title="b1">x2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c15">x1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c16">x2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`1</a> </span>)</span> )</span><br/>

<b>take </b>
<font color="Maroon" title="c16">x2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c15">x1</font>,<font color="Maroon" title="c16">x2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c15">x1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c16">x2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c16">x2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> )</span><br/>

<b>thus </b><a NAME="E3:22_5"/>
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c15">x1</font>,<font color="Maroon" title="c16">x2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c15">x1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c16">x2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c16">x2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="#E2:22_5"><i><font color="Green" title="E28">A18</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c14">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span><b> such that </b><br/><a NAME="E24:22"/><i><font color="Green" title="E28">A19</font></i>: 
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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c14">dx</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E22:22"><i><font color="Green" title="E27">A17</font></i></a>);<br/></i>
<b>defpred </b><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> ] <b>means </b> 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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Olive" title="b1">y1</font>,<font color="Olive" title="b2">y2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`2</a> </span>)</span> );<br/>
<a NAME="E25:22"/><i><font color="Green" title="E29">A20</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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="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#K4" title="BORSUK_1:func.4">:]</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_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">r</font> 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="c15">x</font>,<font color="Olive" title="b1">r</font>]</span><br/>

<b>consider </b><font color="Maroon" title="c16">x1</font>, <font color="Maroon" title="c17">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><b> such that </b><br/><a NAME="E2:22_6"/><i><font color="Green" title="E30">A21</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c16">x1</font>,<font color="Maroon" title="c17">x2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50" title="BORSUK_1:th.50">BORSUK_1:50</a></i>;<br/>
<b>take </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">x1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c17">x2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c15">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">x1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c17">x2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>]</span><br/>

<b>take </b>
<font color="Maroon" title="c16">x1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">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="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c16">x1</font>,<font color="Olive" title="b1">y2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c16">x1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c17">x2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`2</a> </span>)</span> )</span><br/>

<b>take </b>
<font color="Maroon" title="c17">x2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c16">x1</font>,<font color="Maroon" title="c17">x2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c16">x1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c17">x2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c17">x2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> )</span><br/>

<b>thus </b><a NAME="E3:22_6"/>
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c16">x1</font>,<font color="Maroon" title="c17">x2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c16">x1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c17">x2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c17">x2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="#E2:22_6"><i><font color="Green" title="E30">A21</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c15">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span><b> such that </b><br/><a NAME="E27:22"/><i><font color="Green" title="E30">A22</font></i>: 
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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c15">dy</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E25:22"><i><font color="Green" title="E29">A20</font></i></a>);<br/></i>
<b>defpred </b><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> ] <b>means </b> 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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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#K21" title="EUCLID:func.21">`1</a>  );<br/>
<a NAME="E28:22"/><i><font color="Green" title="E31">A23</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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="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#K4" title="BORSUK_1:func.4">:]</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"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">r</font> 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="c16">x</font>,<font color="Olive" title="b1">r</font>]</span><br/>

<b>consider </b><font color="Maroon" title="c17">x1</font>, <font color="Maroon" title="c18">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><b> such that </b><br/><a NAME="E2:22_7"/><i><font color="Green" title="E32">A24</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c17">x1</font>,<font color="Maroon" title="c18">x2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50" title="BORSUK_1:th.50">BORSUK_1:50</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c18">x2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c16">x</font>,<font color="Maroon" title="c18">x2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> ]</span><br/>

<b>take </b>
<font color="Maroon" title="c17">x1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">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="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c17">x1</font>,<font color="Olive" title="b1">x2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> &amp; <font color="Maroon" title="c18">x2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )</span><br/>

<b>take </b>
<font color="Maroon" title="c18">x2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c17">x1</font>,<font color="Maroon" title="c18">x2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> &amp; <font color="Maroon" title="c18">x2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )</span><br/>

<b>thus </b><a NAME="E3:22_7"/>
( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c17">x1</font>,<font color="Maroon" title="c18">x2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> &amp; <font color="Maroon" title="c18">x2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="#E2:22_7"><i><font color="Green" title="E32">A24</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c16">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span><b> such that </b><br/><a NAME="E30:22"/><i><font color="Green" title="E32">A25</font></i>: 
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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> holds  <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c16">fx2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E28:22"><i><font color="Green" title="E31">A23</font></i></a>);<br/></i>
<b>defpred </b><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> ] <b>means </b> 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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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#K22" title="EUCLID:func.22">`2</a>  );<br/>
<a NAME="E31:22"/><i><font color="Green" title="E33">A26</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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="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#K4" title="BORSUK_1:func.4">:]</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_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">r</font> 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="c17">x</font>,<font color="Olive" title="b1">r</font>]</span><br/>

<b>consider </b><font color="Maroon" title="c18">x1</font>, <font color="Maroon" title="c19">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><b> such that </b><br/><a NAME="E2:22_8"/><i><font color="Green" title="E34">A27</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c18">x1</font>,<font color="Maroon" title="c19">x2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50" title="BORSUK_1:th.50">BORSUK_1:50</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c19">x2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>6</sub></font>[<font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c19">x2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> ]</span><br/>

<b>take </b>
<font color="Maroon" title="c18">x1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">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="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c18">x1</font>,<font color="Olive" title="b1">x2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> &amp; <font color="Maroon" title="c19">x2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )</span><br/>

<b>take </b>
<font color="Maroon" title="c19">x2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c18">x1</font>,<font color="Maroon" title="c19">x2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> &amp; <font color="Maroon" title="c19">x2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )</span><br/>

<b>thus </b><a NAME="E3:22_8"/>
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c18">x1</font>,<font color="Maroon" title="c19">x2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> &amp; <font color="Maroon" title="c19">x2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="#E2:22_8"><i><font color="Green" title="E34">A27</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c17">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span><b> such that </b><br/><a NAME="E33:22"/><i><font color="Green" title="E34">A28</font></i>: 
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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> holds  <font color="Maroon">S<sub>6</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c17">fy2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E31:22"><i><font color="Green" title="E33">A26</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c18">Dx</font> = <font color="Maroon" title="c14">dx</font>, <font color="Maroon" title="c19">Dy</font> = <font color="Maroon" title="c15">dy</font>, <font color="Maroon" title="c20">fX2</font> = <font color="Maroon" title="c16">fx2</font>, <font color="Maroon" title="c21">fY2</font> = <font color="Maroon" title="c17">fy2</font>, <font color="Maroon" title="c22">Xo</font> = <font color="Maroon" title="c12">xo</font>, <font color="Maroon" title="c23">Yo</font> = <font color="Maroon" title="c13">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E35:22"/>
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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</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">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</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">Xo</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c24">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">Xo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> st <br/>( <font color="Maroon" title="c24">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">Xo</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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/><b>let </b><font color="Maroon" title="c25">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c22">Xo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">V</font> &amp; <font color="Maroon" title="c25">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> st <br/>( <font color="Maroon" title="c24">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">Xo</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c25">V</font> ) )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:22_9"/><i><font color="Green" title="E35">A29</font></i>: 
<font color="Maroon" title="c22">Xo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">V</font>
 <b>and </b><br/><a NAME="E2:22_9"/><i><font color="Green" title="E36">A30</font></i>: 
<font color="Maroon" title="c25">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> st <br/>( <font color="Maroon" title="c24">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">Xo</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c25">V</font> )</span><br/>

<b>consider </b><font color="Maroon" title="c26">p1</font>, <font color="Maroon" title="c27">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><b> such that </b><br/><a NAME="E4:22_9"/><i><font color="Green" title="E37">A31</font></i>: 
<font color="Maroon" title="c24">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c26">p1</font>,<font color="Maroon" title="c27">p2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>and </b><br/><a NAME="E5:22_9"/><i><font color="Green" title="E38">A32</font></i>: 
<font color="Maroon" title="c22">Xo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E18:22"><i><font color="Green" title="E24">A13</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c28">r</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c27">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c29">V1</font> = <font color="Maroon" title="c25">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>  <b>by </b><i><a class="txt" href="#E2:22_9"><i><font color="Green" title="E36">A30</font></i></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></i>;<br/>
<b>consider </b><font color="Maroon" title="c30">g</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E8:22_9"/><i><font color="Green" title="E39">A33</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c30">g</font>
 <b>and </b><br/><a NAME="E9:22_9"/><i><font color="Green" title="E40">A34</font></i>: 
<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="c27">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c30">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="c27">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c30">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="c29">V1</font>
 <b>by </b><i><a class="txt" href="#E1:22_9"><i><font color="Green" title="E35">A29</font></i></a>, <a class="txt" href="#E5:22_9"><i><font color="Green" title="E38">A32</font></i></a>, <a class="ref" href="rcomp_1.html#T40" title="RCOMP_1:th.40">RCOMP_1:40</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c31">g</font> = <font color="Maroon" title="c30">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>  <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c32">W2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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="c27">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c31">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="c27">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c31">g</font> ) </span>}</span> ;<br/>
<a NAME="E11:22_9"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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="c27">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c31">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="c27">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c31">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"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c33">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c33">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#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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="c27">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c31">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="c27">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c31">g</font> ) </span>}</span>  or <font color="Maroon" title="c33">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/>

<b>assume </b><a NAME="E1:22_9_1"/>
<font color="Maroon" title="c33">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#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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="c27">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c31">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="c27">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c31">g</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c33">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"/><b>then </b>
 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="c33">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c27">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c31">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="c27">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c31">g</font> )
 
;<br/>
<b>hence </b><a NAME="E3:22_9_1"/>
<font color="Maroon" title="c33">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c33">W2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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="c27">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c31">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="c27">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c31">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/>
<b>take </b>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</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="c33">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c24">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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="c33">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> &amp; <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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="c33">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c22">Xo</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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="c33">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c25">V</font> )</span><br/>

<a NAME="E13:22_9"/><i><font color="Green" title="E41">A35</font></i>: 
<font color="Maroon" title="c27">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c27">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/>
<a NAME="E14:22_9"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c27">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c31">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="c27">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c27">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c27">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c31">g</font> )
 
<b>by </b><i><a class="txt" href="#E8:22_9"><i><font color="Green" title="E39">A33</font></i></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></i>;<br/>
<a NAME="E15:22_9"/><b>then </b>
<font color="Maroon" title="c27">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c33">W2</font>
 
<b>by </b><i><a class="txt" href="#E13:22_9"><i><font color="Green" title="E41">A35</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:22_9"/>
<font color="Maroon" title="c24">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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="c33">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E4:22_9"><i><font color="Green" title="E37">A31</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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="c33">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c22">Xo</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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="c33">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c25">V</font> )</span><br/>

<a NAME="E17:22_9"/>
<font color="Maroon" title="c33">W2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T66" title="PSCOMP_1:th.66">PSCOMP_1:66</a></i>;<br/>
<b>hence </b><a NAME="E18:22_9"/>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</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="c33">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T46" title="BORSUK_1:th.46">BORSUK_1:46</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c22">Xo</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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="c33">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c25">V</font></span><br/>

<b>let </b><font color="Maroon" title="c34">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c34">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">Xo</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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="c33">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> or <font color="Maroon" title="c34">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">V</font> )</span><br/>

<b>assume </b><a NAME="E19:22_9"/>
<font color="Maroon" title="c34">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">Xo</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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="c33">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c34">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">V</font></span><br/>

<b>then consider </b><font color="Maroon" title="c35">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span><b> such that </b><br/><a NAME="E21:22_9"/><i><font color="Green" title="E42">A36</font></i>: 
<font color="Maroon" title="c35">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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="c33">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>and </b><br/><a NAME="E22:22_9"/><i><font color="Green" title="E43">A37</font></i>: 
<font color="Maroon" title="c22">Xo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c35">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">b</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c36">a1</font>, <font color="Maroon" title="c37">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><b> such that </b><br/><a NAME="E24:22_9"/><i><font color="Green" title="E44">A38</font></i>: 
<font color="Maroon" title="c35">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c36">a1</font>,<font color="Maroon" title="c37">a2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>and </b><br/><a NAME="E25:22_9"/><i><font color="Green" title="E45">A39</font></i>: 
<font color="Maroon" title="c12">xo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c35">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">a2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E18:22"><i><font color="Green" title="E24">A13</font></i></a></i>;<br/>
<a NAME="E26:22_9"/>
<font color="Maroon" title="c37">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c33">W2</font>
 
<b>by </b><i><a class="txt" href="#E21:22_9"><i><font color="Green" title="E42">A36</font></i></a>, <a class="txt" href="#E24:22_9"><i><font color="Green" title="E44">A38</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c38">x2</font>, <font color="Maroon" title="c39">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> <b> such that </b><br/><a NAME="E28:22_9"/><i><font color="Green" title="E46">A40</font></i>: 
<font color="Maroon" title="c37">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c38">x2</font>,<font color="Maroon" title="c39">y2</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>and </b><br/><a NAME="E29:22_9"/><i><font color="Green" title="E47">A41</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c27">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c31">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c38">x2</font> &amp; <font color="Maroon" title="c38">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="c27">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c31">g</font> )
 ;<br/>
<a NAME="E30:22_9"/>
<font color="Maroon" title="c37">a2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">x2</font>
 
<b>by </b><i><a class="txt" href="#E28:22_9"><i><font color="Green" title="E46">A40</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E31:22_9"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c31">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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c37">a2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c37">a2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c27">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c31">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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="#E29:22_9"><i><font color="Green" title="E47">A41</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E32:22_9"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c37">a2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c27">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c31">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="c27">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c31">g</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></i>;<br/>
<b>hence </b><a NAME="E33:22_9"/>
<font color="Maroon" title="c34">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">V</font>
 <b>by </b><i><a class="txt" href="#E9:22_9"><i><font color="Green" title="E40">A34</font></i></a>, <a class="txt" href="#E22:22_9"><i><font color="Green" title="E43">A37</font></i></a>, <a class="txt" href="#E25:22_9"><i><font color="Green" title="E45">A39</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E36:22"/><b>then </b>
<font color="Maroon" title="c22">Xo</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c24">xo</font> = <font color="Maroon" title="c12">xo</font> as  <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a> <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <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="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#K4" title="BORSUK_1:func.4">:]</a></span> <b>by </b><i><a class="ref" href="topreal6.html#T83" title="TOPREAL6:th.83">TOPREAL6:83</a></i>;<br/>
<a NAME="E38:22"/>
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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</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">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</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">Yo</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c25">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">Yo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> st <br/>( <font color="Maroon" title="c25">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">Yo</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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/><b>let </b><font color="Maroon" title="c26">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c23">Yo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">V</font> &amp; <font color="Maroon" title="c26">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> st <br/>( <font color="Maroon" title="c25">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">Yo</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c26">V</font> ) )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:22_10"/><i><font color="Green" title="E35">A42</font></i>: 
<font color="Maroon" title="c23">Yo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">V</font>
 <b>and </b><br/><a NAME="E2:22_10"/><i><font color="Green" title="E36">A43</font></i>: 
<font color="Maroon" title="c26">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> st <br/>( <font color="Maroon" title="c25">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">Yo</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c26">V</font> )</span><br/>

<b>consider </b><font color="Maroon" title="c27">p1</font>, <font color="Maroon" title="c28">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><b> such that </b><br/><a NAME="E4:22_10"/><i><font color="Green" title="E37">A44</font></i>: 
<font color="Maroon" title="c25">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c27">p1</font>,<font color="Maroon" title="c28">p2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>and </b><br/><a NAME="E5:22_10"/><i><font color="Green" title="E38">A45</font></i>: 
<font color="Maroon" title="c23">Yo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E21:22"><i><font color="Green" title="E26">A16</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c29">r</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c28">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c30">V1</font> = <font color="Maroon" title="c26">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>  <b>by </b><i><a class="txt" href="#E2:22_10"><i><font color="Green" title="E36">A43</font></i></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></i>;<br/>
<b>consider </b><font color="Maroon" title="c31">g</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E8:22_10"/><i><font color="Green" title="E39">A46</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c31">g</font>
 <b>and </b><br/><a NAME="E9:22_10"/><i><font color="Green" title="E40">A47</font></i>: 
<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="c28">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c31">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="c28">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c31">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>
 <b>by </b><i><a class="txt" href="#E1:22_10"><i><font color="Green" title="E35">A42</font></i></a>, <a class="txt" href="#E5:22_10"><i><font color="Green" title="E38">A45</font></i></a>, <a class="ref" href="rcomp_1.html#T40" title="RCOMP_1:th.40">RCOMP_1:40</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c32">g</font> = <font color="Maroon" title="c31">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>  <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c33">W2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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="c28">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c32">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="c28">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c32">g</font> ) </span>}</span> ;<br/>
<a NAME="E11:22_10"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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="c28">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c32">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="c28">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c32">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"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c34">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c34">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#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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="c28">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c32">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="c28">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c32">g</font> ) </span>}</span>  or <font color="Maroon" title="c34">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/>

<b>assume </b><a NAME="E1:22_10_1"/>
<font color="Maroon" title="c34">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#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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="c28">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c32">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="c28">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c32">g</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c34">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"/><b>then </b>
 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="c34">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c28">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c32">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="c28">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c32">g</font> )
 
;<br/>
<b>hence </b><a NAME="E3:22_10_1"/>
<font color="Maroon" title="c34">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c34">W2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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="c28">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c32">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="c28">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c32">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/>
<b>take </b>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</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="c34">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c25">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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="c34">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> &amp; <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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="c34">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c23">Yo</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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="c34">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c26">V</font> )</span><br/>

<a NAME="E13:22_10"/><i><font color="Green" title="E41">A48</font></i>: 
<font color="Maroon" title="c28">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/>
<a NAME="E14:22_10"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c28">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c32">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="c28">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c28">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c28">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c32">g</font> )
 
<b>by </b><i><a class="txt" href="#E8:22_10"><i><font color="Green" title="E39">A46</font></i></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></i>;<br/>
<a NAME="E15:22_10"/><b>then </b>
<font color="Maroon" title="c28">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">W2</font>
 
<b>by </b><i><a class="txt" href="#E13:22_10"><i><font color="Green" title="E41">A48</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:22_10"/>
<font color="Maroon" title="c25">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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="c34">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E4:22_10"><i><font color="Green" title="E37">A44</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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="c34">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c23">Yo</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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="c34">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c26">V</font> )</span><br/>

<a NAME="E17:22_10"/>
<font color="Maroon" title="c34">W2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T68" title="PSCOMP_1:th.68">PSCOMP_1:68</a></i>;<br/>
<b>hence </b><a NAME="E18:22_10"/>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</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="c34">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T46" title="BORSUK_1:th.46">BORSUK_1:46</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c23">Yo</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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="c34">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c26">V</font></span><br/>

<b>let </b><font color="Maroon" title="c35">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c35">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">Yo</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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="c34">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> or <font color="Maroon" title="c35">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">V</font> )</span><br/>

<b>assume </b><a NAME="E19:22_10"/>
<font color="Maroon" title="c35">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">Yo</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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="c34">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c35">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">V</font></span><br/>

<b>then consider </b><font color="Maroon" title="c36">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span><b> such that </b><br/><a NAME="E21:22_10"/><i><font color="Green" title="E42">A49</font></i>: 
<font color="Maroon" title="c36">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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="c34">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>and </b><br/><a NAME="E22:22_10"/><i><font color="Green" title="E43">A50</font></i>: 
<font color="Maroon" title="c23">Yo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c36">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">b</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c37">a1</font>, <font color="Maroon" title="c38">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><b> such that </b><br/><a NAME="E24:22_10"/><i><font color="Green" title="E44">A51</font></i>: 
<font color="Maroon" title="c36">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c37">a1</font>,<font color="Maroon" title="c38">a2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>and </b><br/><a NAME="E25:22_10"/><i><font color="Green" title="E45">A52</font></i>: 
<font color="Maroon" title="c13">yo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c36">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">a2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E21:22"><i><font color="Green" title="E26">A16</font></i></a></i>;<br/>
<a NAME="E26:22_10"/>
<font color="Maroon" title="c38">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">W2</font>
 
<b>by </b><i><a class="txt" href="#E21:22_10"><i><font color="Green" title="E42">A49</font></i></a>, <a class="txt" href="#E24:22_10"><i><font color="Green" title="E44">A51</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c39">x2</font>, <font color="Maroon" title="c40">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> <b> such that </b><br/><a NAME="E28:22_10"/><i><font color="Green" title="E46">A53</font></i>: 
<font color="Maroon" title="c38">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c39">x2</font>,<font color="Maroon" title="c40">y2</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>and </b><br/><a NAME="E29:22_10"/><i><font color="Green" title="E47">A54</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c28">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c32">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c40">y2</font> &amp; <font color="Maroon" title="c40">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="c28">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c32">g</font> )
 ;<br/>
<a NAME="E30:22_10"/>
<font color="Maroon" title="c38">a2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c40">y2</font>
 
<b>by </b><i><a class="txt" href="#E28:22_10"><i><font color="Green" title="E46">A53</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E31:22_10"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c32">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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c38">a2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c38">a2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c28">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c32">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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="#E29:22_10"><i><font color="Green" title="E47">A54</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E32:22_10"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c38">a2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c28">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c32">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="c28">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c32">g</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></i>;<br/>
<b>hence </b><a NAME="E33:22_10"/>
<font color="Maroon" title="c35">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">V</font>
 <b>by </b><i><a class="txt" href="#E9:22_10"><i><font color="Green" title="E40">A47</font></i></a>, <a class="txt" href="#E22:22_10"><i><font color="Green" title="E43">A50</font></i></a>, <a class="txt" href="#E25:22_10"><i><font color="Green" title="E45">A52</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E39:22"/><b>then </b>
<font color="Maroon" title="c23">Yo</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c25">yo</font> = <font color="Maroon" title="c13">yo</font> as  <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a> <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <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="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#K4" title="BORSUK_1:func.4">:]</a></span> <b>by </b><i><a class="ref" href="topreal6.html#T83" title="TOPREAL6:th.83">TOPREAL6:83</a></i>;<br/>
<a NAME="E41:22"/>
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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</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="c18">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</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="c18">Dx</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c26">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c18">Dx</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> st <br/>( <font color="Maroon" title="c26">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="c18">Dx</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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/><b>let </b><font color="Maroon" title="c27">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c18">Dx</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">V</font> &amp; <font color="Maroon" title="c27">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> st <br/>( <font color="Maroon" title="c26">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="c18">Dx</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c27">V</font> ) )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:22_11"/><i><font color="Green" title="E35">A55</font></i>: 
<font color="Maroon" title="c18">Dx</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">V</font>
 <b>and </b><br/><a NAME="E2:22_11"/><i><font color="Green" title="E36">A56</font></i>: 
<font color="Maroon" title="c27">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> st <br/>( <font color="Maroon" title="c26">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="c18">Dx</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c27">V</font> )</span><br/>

<b>consider </b><font color="Maroon" title="c28">p1</font>, <font color="Maroon" title="c29">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><b> such that </b><br/><a NAME="E4:22_11"/><i><font color="Green" title="E37">A57</font></i>: 
<font color="Maroon" title="c26">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c28">p1</font>,<font color="Maroon" title="c29">p2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>and </b><br/><a NAME="E5:22_11"/><i><font color="Green" title="E38">A58</font></i>: 
<font color="Maroon" title="c14">dx</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E24:22"><i><font color="Green" title="E28">A19</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c30">r</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c31">V1</font> = <font color="Maroon" title="c27">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>  <b>by </b><i><a class="txt" href="#E2:22_11"><i><font color="Green" title="E36">A56</font></i></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></i>;<br/>
<b>consider </b><font color="Maroon" title="c32">g</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E8:22_11"/><i><font color="Green" title="E39">A59</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c32">g</font>
 <b>and </b><br/><a NAME="E9:22_11"/><i><font color="Green" title="E40">A60</font></i>: 
<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="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c32">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="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c32">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>
 <b>by </b><i><a class="txt" href="#E1:22_11"><i><font color="Green" title="E35">A55</font></i></a>, <a class="txt" href="#E5:22_11"><i><font color="Green" title="E38">A58</font></i></a>, <a class="ref" href="rcomp_1.html#T40" title="RCOMP_1:th.40">RCOMP_1:40</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c33">g</font> = <font color="Maroon" title="c32">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>  <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c34">W1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">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="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c35">W2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">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="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">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="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">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="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">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"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c36">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c36">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#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">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="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> ) </span>}</span>  or <font color="Maroon" title="c36">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/>

<b>assume </b><a NAME="E1:22_11_1"/>
<font color="Maroon" title="c36">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#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">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="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c36">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"/><b>then </b>
 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="c36">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">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="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E3:22_11_1"/>
<font color="Maroon" title="c36">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c36">W1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">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="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">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/>
<a NAME="E13:22_11"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">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="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">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"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c37">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="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#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">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="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> ) </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/>

<b>assume </b><a NAME="E1:22_11_2"/>
<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#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">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="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">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:()"><i><font color="Red">::  thesis: </font></i></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_11_2"/><b>then </b>
 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#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">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="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E3:22_11_2"/>
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c37">W2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">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="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">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/>
<b>take </b>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c36">W1</font>,<font color="Maroon" title="c37">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c26">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c36">W1</font>,<font color="Maroon" title="c37">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> &amp; <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c36">W1</font>,<font color="Maroon" title="c37">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c18">Dx</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c36">W1</font>,<font color="Maroon" title="c37">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c27">V</font> )</span><br/>

<a NAME="E15:22_11"/><i><font color="Green" title="E41">A61</font></i>: 
<font color="Maroon" title="c28">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/>
<a NAME="E16:22_11"/><i><font color="Green" title="E42">A62</font></i>: 
<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="c33">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<b>by </b><i><a class="txt" href="#E8:22_11"><i><font color="Green" title="E39">A59</font></i></a>, <a class="ref" href="xreal_1.html#T76" title="XREAL_1:th.76">XREAL_1:76</a></i>;<br/>
<a NAME="E17:22_11"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">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="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> )
 
<b>by </b><i><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></i>;<br/>
<a NAME="E18:22_11"/><b>then </b><i><font color="Green" title="E43">A63</font></i>: 
<font color="Maroon" title="c28">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c36">W1</font>
 
<b>by </b><i><a class="txt" href="#E15:22_11"><i><font color="Green" title="E41">A61</font></i></a></i>;<br/>
<a NAME="E19:22_11"/><i><font color="Green" title="E44">A64</font></i>: 
<font color="Maroon" title="c29">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c29">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/>
<a NAME="E20:22_11"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">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="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> )
 
<b>by </b><i><a class="txt" href="#E16:22_11"><i><font color="Green" title="E42">A62</font></i></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></i>;<br/>
<a NAME="E21:22_11"/><b>then </b>
<font color="Maroon" title="c29">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c37">W2</font>
 
<b>by </b><i><a class="txt" href="#E19:22_11"><i><font color="Green" title="E44">A64</font></i></a></i>;<br/>
<b>hence </b><a NAME="E22:22_11"/>
<font color="Maroon" title="c26">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c36">W1</font>,<font color="Maroon" title="c37">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E4:22_11"><i><font color="Green" title="E37">A57</font></i></a>, <a class="txt" href="#E18:22_11"><i><font color="Green" title="E43">A63</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c36">W1</font>,<font color="Maroon" title="c37">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c18">Dx</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c36">W1</font>,<font color="Maroon" title="c37">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c27">V</font> )</span><br/>

<a NAME="E23:22_11"/>
( <font color="Maroon" title="c36">W1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c37">W2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T66" title="PSCOMP_1:th.66">PSCOMP_1:66</a></i>;<br/>
<b>hence </b><a NAME="E24:22_11"/>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c36">W1</font>,<font color="Maroon" title="c37">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T46" title="BORSUK_1:th.46">BORSUK_1:46</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c18">Dx</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c36">W1</font>,<font color="Maroon" title="c37">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c27">V</font></span><br/>

<b>let </b><font color="Maroon" title="c38">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c38">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Dx</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c36">W1</font>,<font color="Maroon" title="c37">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</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="c27">V</font> )</span><br/>

<b>assume </b><a NAME="E25:22_11"/>
<font color="Maroon" title="c38">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Dx</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c36">W1</font>,<font color="Maroon" title="c37">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c38">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">V</font></span><br/>

<b>then consider </b><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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span><b> such that </b><br/><a NAME="E27:22_11"/><i><font color="Green" title="E45">A65</font></i>: 
<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#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c36">W1</font>,<font color="Maroon" title="c37">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>and </b><br/><a NAME="E28:22_11"/><i><font color="Green" title="E46">A66</font></i>: 
<font color="Maroon" title="c18">Dx</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>
 <b>by </b><i><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<b>consider </b><font color="Maroon" title="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><b> such that </b><br/><a NAME="E30:22_11"/><i><font color="Green" title="E47">A67</font></i>: 
<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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c40">a1</font>,<font color="Maroon" title="c41">a2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>and </b><br/><a NAME="E31:22_11"/><i><font color="Green" title="E48">A68</font></i>: 
<font color="Maroon" title="c14">dx</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="c40">a1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c41">a2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E24:22"><i><font color="Green" title="E28">A19</font></i></a></i>;<br/>
<a NAME="E32:22_11"/>
<font color="Maroon" title="c40">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c36">W1</font>
 
<b>by </b><i><a class="txt" href="#E27:22_11"><i><font color="Green" title="E45">A65</font></i></a>, <a class="txt" href="#E30:22_11"><i><font color="Green" title="E47">A67</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c42">x1</font>, <font color="Maroon" title="c43">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> <b> such that </b><br/><a NAME="E34:22_11"/><i><font color="Green" title="E49">A69</font></i>: 
<font color="Maroon" title="c40">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c42">x1</font>,<font color="Maroon" title="c43">y1</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>and </b><br/><a NAME="E35:22_11"/><i><font color="Green" title="E50">A70</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">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="c42">x1</font>
 <b>and </b><br/><a NAME="E36:22_11"/><i><font color="Green" title="E51">A71</font></i>: 
<font color="Maroon" title="c42">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="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 ;<br/>
<a NAME="E37:22_11"/><i><font color="Green" title="E52">A72</font></i>: 
<font color="Maroon" title="c40">a1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">x1</font>
 
<b>by </b><i><a class="txt" href="#E34:22_11"><i><font color="Green" title="E49">A69</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E38:22_11"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">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="c33">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="c42">x1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c42">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="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E35:22_11"><i><font color="Green" title="E50">A70</font></i></a>, <a class="txt" href="#E36:22_11"><i><font color="Green" title="E51">A71</font></i></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></i>;<br/>
<a NAME="E39:22_11"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c42">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="c42">x1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c33">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="c42">x1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c42">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="c33">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E40:22_11"/><b>then </b><i><font color="Green" title="E53">A73</font></i>: 
 <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="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c42">x1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c33">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<b>by </b><i><a class="ref" href="seq_2.html#T9" title="SEQ_2:th.9">SEQ_2:9</a></i>;<br/>
<a NAME="E41:22_11"/>
<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>
 
<b>by </b><i><a class="txt" href="#E27:22_11"><i><font color="Green" title="E45">A65</font></i></a>, <a class="txt" href="#E30:22_11"><i><font color="Green" title="E47">A67</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c44">x2</font>, <font color="Maroon" title="c45">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> <b> such that </b><br/><a NAME="E43:22_11"/><i><font color="Green" title="E54">A74</font></i>: 
<font color="Maroon" title="c41">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c44">x2</font>,<font color="Maroon" title="c45">y2</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>and </b><br/><a NAME="E44:22_11"/><i><font color="Green" title="E55">A75</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">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="c44">x2</font>
 <b>and </b><br/><a NAME="E45:22_11"/><i><font color="Green" title="E56">A76</font></i>: 
<font color="Maroon" title="c44">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="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 ;<br/>
<a NAME="E46:22_11"/><i><font color="Green" title="E57">A77</font></i>: 
<font color="Maroon" title="c41">a2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">x2</font>
 
<b>by </b><i><a class="txt" href="#E43:22_11"><i><font color="Green" title="E54">A74</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E47:22_11"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">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="c33">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="c44">x2</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c44">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="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E44:22_11"><i><font color="Green" title="E55">A75</font></i></a>, <a class="txt" href="#E45:22_11"><i><font color="Green" title="E56">A76</font></i></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></i>;<br/>
<a NAME="E48:22_11"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c44">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="c44">x2</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c33">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="c44">x2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c44">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="c33">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E49:22_11"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c44">x2</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c33">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<b>by </b><i><a class="ref" href="seq_2.html#T9" title="SEQ_2:th.9">SEQ_2:9</a></i>;<br/>
<a NAME="E50:22_11"/><b>then </b><i><font color="Green" title="E58">A78</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c42">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="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c44">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="c33">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="c33">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="#E40:22_11"><i><font color="Green" title="E53">A73</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E51:22_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="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c42">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="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c44">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="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c42">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="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c44">x2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T143" title="COMPLEX1:th.143">COMPLEX1:143</a></i>;<br/>
<a NAME="E52:22_11"/><b>then </b>
 <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="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c42">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="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c44">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="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c42">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="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c44">x2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></i>;<br/>
<a NAME="E53:22_11"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c42">x1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c44">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="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c33">g</font>
 
<b>by </b><i><a class="txt" href="#E50:22_11"><i><font color="Green" title="E58">A78</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E54:22_11"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c40">a1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c41">a2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c33">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="c28">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c29">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c33">g</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="txt" href="#E37:22_11"><i><font color="Green" title="E52">A72</font></i></a>, <a class="txt" href="#E46:22_11"><i><font color="Green" title="E57">A77</font></i></a>, <a class="ref" href="rcomp_1.html#T8" title="RCOMP_1:th.8">RCOMP_1:8</a></i>;<br/>
<b>hence </b><a NAME="E55:22_11"/>
<font color="Maroon" title="c38">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">V</font>
 <b>by </b><i><a class="txt" href="#E9:22_11"><i><font color="Green" title="E40">A60</font></i></a>, <a class="txt" href="#E28:22_11"><i><font color="Green" title="E46">A66</font></i></a>, <a class="txt" href="#E31:22_11"><i><font color="Green" title="E48">A68</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E42:22"/><b>then </b>
<font color="Maroon" title="c18">Dx</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c26">dx</font> = <font color="Maroon" title="c14">dx</font> as  <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a> <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <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="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#K4" title="BORSUK_1:func.4">:]</a></span> <b>by </b><i><a class="ref" href="topreal6.html#T83" title="TOPREAL6:th.83">TOPREAL6:83</a></i>;<br/>
<a NAME="E44:22"/>
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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</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="c19">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</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="c19">Dy</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c27">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c19">Dy</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> st <br/>( <font color="Maroon" title="c27">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="c19">Dy</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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/><b>let </b><font color="Maroon" title="c28">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c19">Dy</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">V</font> &amp; <font color="Maroon" title="c28">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> st <br/>( <font color="Maroon" title="c27">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="c19">Dy</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c28">V</font> ) )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:22_12"/><i><font color="Green" title="E35">A79</font></i>: 
<font color="Maroon" title="c19">Dy</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">V</font>
 <b>and </b><br/><a NAME="E2:22_12"/><i><font color="Green" title="E36">A80</font></i>: 
<font color="Maroon" title="c28">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> st <br/>( <font color="Maroon" title="c27">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="c19">Dy</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c28">V</font> )</span><br/>

<b>consider </b><font color="Maroon" title="c29">p1</font>, <font color="Maroon" title="c30">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><b> such that </b><br/><a NAME="E4:22_12"/><i><font color="Green" title="E37">A81</font></i>: 
<font color="Maroon" title="c27">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c29">p1</font>,<font color="Maroon" title="c30">p2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>and </b><br/><a NAME="E5:22_12"/><i><font color="Green" title="E38">A82</font></i>: 
<font color="Maroon" title="c15">dy</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E27:22"><i><font color="Green" title="E30">A22</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c31">r</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c32">V1</font> = <font color="Maroon" title="c28">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>  <b>by </b><i><a class="txt" href="#E2:22_12"><i><font color="Green" title="E36">A80</font></i></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></i>;<br/>
<b>consider </b><font color="Maroon" title="c33">g</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E8:22_12"/><i><font color="Green" title="E39">A83</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c33">g</font>
 <b>and </b><br/><a NAME="E9:22_12"/><i><font color="Green" title="E40">A84</font></i>: 
<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="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c33">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="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c33">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="c32">V1</font>
 <b>by </b><i><a class="txt" href="#E1:22_12"><i><font color="Green" title="E35">A79</font></i></a>, <a class="txt" href="#E5:22_12"><i><font color="Green" title="E38">A82</font></i></a>, <a class="ref" href="rcomp_1.html#T40" title="RCOMP_1:th.40">RCOMP_1:40</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c34">g</font> = <font color="Maroon" title="c33">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>  <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c35">W1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">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="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c36">W2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">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="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">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="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">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="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">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"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c37">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="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#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">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="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> ) </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/>

<b>assume </b><a NAME="E1:22_12_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#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">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="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">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:()"><i><font color="Red">::  thesis: </font></i></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_12_1"/><b>then </b>
 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#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">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="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E3:22_12_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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c37">W1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">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="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">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/>
<a NAME="E13:22_12"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">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="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">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"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c38">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="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#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">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="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> ) </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/>

<b>assume </b><a NAME="E1:22_12_2"/>
<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#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">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="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">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:()"><i><font color="Red">::  thesis: </font></i></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_12_2"/><b>then </b>
 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#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">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="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E3:22_12_2"/>
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c38">W2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">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="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">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/>
<b>take </b>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c37">W1</font>,<font color="Maroon" title="c38">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c27">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c37">W1</font>,<font color="Maroon" title="c38">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> &amp; <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c37">W1</font>,<font color="Maroon" title="c38">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c19">Dy</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c37">W1</font>,<font color="Maroon" title="c38">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c28">V</font> )</span><br/>

<a NAME="E15:22_12"/><i><font color="Green" title="E41">A85</font></i>: 
<font color="Maroon" title="c29">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/>
<a NAME="E16:22_12"/><i><font color="Green" title="E42">A86</font></i>: 
<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="c34">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<b>by </b><i><a class="txt" href="#E8:22_12"><i><font color="Green" title="E39">A83</font></i></a>, <a class="ref" href="xreal_1.html#T76" title="XREAL_1:th.76">XREAL_1:76</a></i>;<br/>
<a NAME="E17:22_12"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">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="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> )
 
<b>by </b><i><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></i>;<br/>
<a NAME="E18:22_12"/><b>then </b><i><font color="Green" title="E43">A87</font></i>: 
<font color="Maroon" title="c29">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c37">W1</font>
 
<b>by </b><i><a class="txt" href="#E15:22_12"><i><font color="Green" title="E41">A85</font></i></a></i>;<br/>
<a NAME="E19:22_12"/><i><font color="Green" title="E44">A88</font></i>: 
<font color="Maroon" title="c30">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c30">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/>
<a NAME="E20:22_12"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">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="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> )
 
<b>by </b><i><a class="txt" href="#E16:22_12"><i><font color="Green" title="E42">A86</font></i></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></i>;<br/>
<a NAME="E21:22_12"/><b>then </b>
<font color="Maroon" title="c30">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c38">W2</font>
 
<b>by </b><i><a class="txt" href="#E19:22_12"><i><font color="Green" title="E44">A88</font></i></a></i>;<br/>
<b>hence </b><a NAME="E22:22_12"/>
<font color="Maroon" title="c27">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c37">W1</font>,<font color="Maroon" title="c38">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E4:22_12"><i><font color="Green" title="E37">A81</font></i></a>, <a class="txt" href="#E18:22_12"><i><font color="Green" title="E43">A87</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c37">W1</font>,<font color="Maroon" title="c38">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c19">Dy</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c37">W1</font>,<font color="Maroon" title="c38">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c28">V</font> )</span><br/>

<a NAME="E23:22_12"/>
( <font color="Maroon" title="c37">W1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c38">W2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T68" title="PSCOMP_1:th.68">PSCOMP_1:68</a></i>;<br/>
<b>hence </b><a NAME="E24:22_12"/>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c37">W1</font>,<font color="Maroon" title="c38">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T46" title="BORSUK_1:th.46">BORSUK_1:46</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c19">Dy</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c37">W1</font>,<font color="Maroon" title="c38">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c28">V</font></span><br/>

<b>let </b><font color="Maroon" title="c39">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c39">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">Dy</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c37">W1</font>,<font color="Maroon" title="c38">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</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="c28">V</font> )</span><br/>

<b>assume </b><a NAME="E25:22_12"/>
<font color="Maroon" title="c39">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">Dy</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c37">W1</font>,<font color="Maroon" title="c38">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c39">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">V</font></span><br/>

<b>then consider </b><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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span><b> such that </b><br/><a NAME="E27:22_12"/><i><font color="Green" title="E45">A89</font></i>: 
<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#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c37">W1</font>,<font color="Maroon" title="c38">W2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>and </b><br/><a NAME="E28:22_12"/><i><font color="Green" title="E46">A90</font></i>: 
<font color="Maroon" title="c19">Dy</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>
 <b>by </b><i><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<b>consider </b><font color="Maroon" title="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><b> such that </b><br/><a NAME="E30:22_12"/><i><font color="Green" title="E47">A91</font></i>: 
<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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c41">a1</font>,<font color="Maroon" title="c42">a2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>and </b><br/><a NAME="E31:22_12"/><i><font color="Green" title="E48">A92</font></i>: 
<font color="Maroon" title="c15">dy</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="c41">a1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c42">a2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E27:22"><i><font color="Green" title="E30">A22</font></i></a></i>;<br/>
<a NAME="E32:22_12"/>
<font color="Maroon" title="c41">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c37">W1</font>
 
<b>by </b><i><a class="txt" href="#E27:22_12"><i><font color="Green" title="E45">A89</font></i></a>, <a class="txt" href="#E30:22_12"><i><font color="Green" title="E47">A91</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c43">x1</font>, <font color="Maroon" title="c44">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> <b> such that </b><br/><a NAME="E34:22_12"/><i><font color="Green" title="E49">A93</font></i>: 
<font color="Maroon" title="c41">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c43">x1</font>,<font color="Maroon" title="c44">y1</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>and </b><br/><a NAME="E35:22_12"/><i><font color="Green" title="E50">A94</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">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="c44">y1</font>
 <b>and </b><br/><a NAME="E36:22_12"/><i><font color="Green" title="E51">A95</font></i>: 
<font color="Maroon" title="c44">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="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 ;<br/>
<a NAME="E37:22_12"/><i><font color="Green" title="E52">A96</font></i>: 
<font color="Maroon" title="c41">a1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">y1</font>
 
<b>by </b><i><a class="txt" href="#E34:22_12"><i><font color="Green" title="E49">A93</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E38:22_12"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">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="c34">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="c44">y1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c44">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="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E35:22_12"><i><font color="Green" title="E50">A94</font></i></a>, <a class="txt" href="#E36:22_12"><i><font color="Green" title="E51">A95</font></i></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></i>;<br/>
<a NAME="E39:22_12"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c44">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="c44">y1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">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="c44">y1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c44">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="c34">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E40:22_12"/><b>then </b><i><font color="Green" title="E53">A97</font></i>: 
 <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="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c44">y1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c34">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<b>by </b><i><a class="ref" href="seq_2.html#T9" title="SEQ_2:th.9">SEQ_2:9</a></i>;<br/>
<a NAME="E41:22_12"/>
<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>
 
<b>by </b><i><a class="txt" href="#E27:22_12"><i><font color="Green" title="E45">A89</font></i></a>, <a class="txt" href="#E30:22_12"><i><font color="Green" title="E47">A91</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c45">x2</font>, <font color="Maroon" title="c46">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> <b> such that </b><br/><a NAME="E43:22_12"/><i><font color="Green" title="E54">A98</font></i>: 
<font color="Maroon" title="c42">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c45">x2</font>,<font color="Maroon" title="c46">y2</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>and </b><br/><a NAME="E44:22_12"/><i><font color="Green" title="E55">A99</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">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="c46">y2</font>
 <b>and </b><br/><a NAME="E45:22_12"/><i><font color="Green" title="E56">A100</font></i>: 
<font color="Maroon" title="c46">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="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 ;<br/>
<a NAME="E46:22_12"/><i><font color="Green" title="E57">A101</font></i>: 
<font color="Maroon" title="c42">a2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c46">y2</font>
 
<b>by </b><i><a class="txt" href="#E43:22_12"><i><font color="Green" title="E54">A98</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E47:22_12"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">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="c34">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="c46">y2</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c46">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="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E44:22_12"><i><font color="Green" title="E55">A99</font></i></a>, <a class="txt" href="#E45:22_12"><i><font color="Green" title="E56">A100</font></i></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></i>;<br/>
<a NAME="E48:22_12"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c46">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="c46">y2</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">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="c46">y2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c46">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="c34">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E49:22_12"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c46">y2</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c34">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<b>by </b><i><a class="ref" href="seq_2.html#T9" title="SEQ_2:th.9">SEQ_2:9</a></i>;<br/>
<a NAME="E50:22_12"/><b>then </b><i><font color="Green" title="E58">A102</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c44">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="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c46">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="c34">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="c34">g</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="#E40:22_12"><i><font color="Green" title="E53">A97</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E51:22_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="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c44">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="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c46">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="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c44">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="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c46">y2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T143" title="COMPLEX1:th.143">COMPLEX1:143</a></i>;<br/>
<a NAME="E52:22_12"/><b>then </b>
 <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="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c44">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="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c46">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="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c44">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="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c46">y2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></i>;<br/>
<a NAME="E53:22_12"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c44">y1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c46">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="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c34">g</font>
 
<b>by </b><i><a class="txt" href="#E50:22_12"><i><font color="Green" title="E58">A102</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E54:22_12"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c41">a1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c42">a2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c29">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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>
 
<b>by </b><i><a class="txt" href="#E37:22_12"><i><font color="Green" title="E52">A96</font></i></a>, <a class="txt" href="#E46:22_12"><i><font color="Green" title="E57">A101</font></i></a>, <a class="ref" href="rcomp_1.html#T8" title="RCOMP_1:th.8">RCOMP_1:8</a></i>;<br/>
<b>hence </b><a NAME="E55:22_12"/>
<font color="Maroon" title="c39">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">V</font>
 <b>by </b><i><a class="txt" href="#E9:22_12"><i><font color="Green" title="E40">A84</font></i></a>, <a class="txt" href="#E28:22_12"><i><font color="Green" title="E46">A90</font></i></a>, <a class="txt" href="#E31:22_12"><i><font color="Green" title="E48">A92</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E45:22"/><b>then </b>
<font color="Maroon" title="c19">Dy</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c27">dy</font> = <font color="Maroon" title="c15">dy</font> as  <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a> <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <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="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#K4" title="BORSUK_1:func.4">:]</a></span> <b>by </b><i><a class="ref" href="topreal6.html#T83" title="TOPREAL6:th.83">TOPREAL6:83</a></i>;<br/>
<a NAME="E47:22"/>
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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</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="c20">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</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="c20">fX2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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"><b>proof </b></a><div class="add">

<b>let </b><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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c20">fX2</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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</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="c20">fX2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c20">fX2</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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</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="c20">fX2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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/>



<b>assume </b><b>that </b><br/><a NAME="E1:22_13"/><i><font color="Green" title="E35">A103</font></i>: 
<font color="Maroon" title="c20">fX2</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>
 <b>and </b><br/><a NAME="E2:22_13"/><i><font color="Green" title="E36">A104</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</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="c20">fX2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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/>

<b>consider </b><font color="Maroon" title="c30">p1</font>, <font color="Maroon" title="c31">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><b> such that </b><br/><a NAME="E4:22_13"/><i><font color="Green" title="E37">A105</font></i>: 
<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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c30">p1</font>,<font color="Maroon" title="c31">p2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>and </b><br/><a NAME="E5:22_13"/><i><font color="Green" title="E38">A106</font></i>: 
<font color="Maroon" title="c20">fX2</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> <font color="Maroon" title="c31">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="#E30:22"><i><font color="Green" title="E32">A25</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c32">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>  <b>by </b><i><a class="txt" href="#E2:22_13"><i><font color="Green" title="E36">A104</font></i></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></i>;<br/>
<b>consider </b><font color="Maroon" title="c33">g</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E8:22_13"/><i><font color="Green" title="E39">A107</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c33">g</font>
 <b>and </b><br/><a NAME="E9:22_13"/><i><font color="Green" title="E40">A108</font></i>: 
<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="c31">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c33">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c31">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c33">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="c32">V1</font>
 <b>by </b><i><a class="txt" href="#E1:22_13"><i><font color="Green" title="E35">A103</font></i></a>, <a class="txt" href="#E5:22_13"><i><font color="Green" title="E38">A106</font></i></a>, <a class="ref" href="rcomp_1.html#T40" title="RCOMP_1:th.40">RCOMP_1:40</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c34">g</font> = <font color="Maroon" title="c33">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>  <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c35">W1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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="c31">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c34">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="c31">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c34">g</font> ) </span>}</span> ;<br/>
<a NAME="E11:22_13"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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="c31">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c34">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="c31">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c34">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"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c36">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c36">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#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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="c31">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c34">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="c31">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c34">g</font> ) </span>}</span>  or <font color="Maroon" title="c36">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/>

<b>assume </b><a NAME="E1:22_13_1"/>
<font color="Maroon" title="c36">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#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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="c31">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c34">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="c31">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c34">g</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c36">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"/><b>then </b>
 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="c36">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c31">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c34">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="c31">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c34">g</font> )
 
;<br/>
<b>hence </b><a NAME="E3:22_13_1"/>
<font color="Maroon" title="c36">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c36">W1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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="c31">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c34">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="c31">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c34">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/>
<b>take </b>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</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="c36">W1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c28">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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="c36">W1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> &amp; <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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="c36">W1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c20">fX2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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="c36">W1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c29">V</font> )</span><br/>

<a NAME="E13:22_13"/><i><font color="Green" title="E41">A109</font></i>: 
<font color="Maroon" title="c31">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c31">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c31">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/>
<a NAME="E14:22_13"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c31">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c34">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="c31">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c31">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c31">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c34">g</font> )
 
<b>by </b><i><a class="txt" href="#E8:22_13"><i><font color="Green" title="E39">A107</font></i></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></i>;<br/>
<a NAME="E15:22_13"/><b>then </b>
<font color="Maroon" title="c31">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c36">W1</font>
 
<b>by </b><i><a class="txt" href="#E13:22_13"><i><font color="Green" title="E41">A109</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:22_13"/>
<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#K5" title="BORSUK_1:func.5">[:</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="c36">W1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E4:22_13"><i><font color="Green" title="E37">A105</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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="c36">W1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c20">fX2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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="c36">W1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c29">V</font> )</span><br/>

<a NAME="E17:22_13"/>
<font color="Maroon" title="c36">W1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T66" title="PSCOMP_1:th.66">PSCOMP_1:66</a></i>;<br/>
<b>hence </b><a NAME="E18:22_13"/>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</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="c36">W1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T46" title="BORSUK_1:th.46">BORSUK_1:46</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c20">fX2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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="c36">W1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c29">V</font></span><br/>

<b>let </b><font color="Maroon" title="c37">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c37">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">fX2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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="c36">W1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> or <font color="Maroon" title="c37">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">V</font> )</span><br/>

<b>assume </b><a NAME="E19:22_13"/>
<font color="Maroon" title="c37">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">fX2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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="c36">W1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c37">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">V</font></span><br/>

<b>then consider </b><font color="Maroon" title="c38">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span><b> such that </b><br/><a NAME="E21:22_13"/><i><font color="Green" title="E42">A110</font></i>: 
<font color="Maroon" title="c38">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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="c36">W1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>and </b><br/><a NAME="E22:22_13"/><i><font color="Green" title="E43">A111</font></i>: 
<font color="Maroon" title="c20">fX2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c38">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">b</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c39">a1</font>, <font color="Maroon" title="c40">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><b> such that </b><br/><a NAME="E24:22_13"/><i><font color="Green" title="E44">A112</font></i>: 
<font color="Maroon" title="c38">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c39">a1</font>,<font color="Maroon" title="c40">a2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>and </b><br/><a NAME="E25:22_13"/><i><font color="Green" title="E45">A113</font></i>: 
<font color="Maroon" title="c20">fX2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c38">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c40">a2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="#E30:22"><i><font color="Green" title="E32">A25</font></i></a></i>;<br/>
<a NAME="E26:22_13"/>
<font color="Maroon" title="c40">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c36">W1</font>
 
<b>by </b><i><a class="txt" href="#E21:22_13"><i><font color="Green" title="E42">A110</font></i></a>, <a class="txt" href="#E24:22_13"><i><font color="Green" title="E44">A112</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c41">x1</font>, <font color="Maroon" title="c42">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> <b> such that </b><br/><a NAME="E28:22_13"/><i><font color="Green" title="E46">A114</font></i>: 
<font color="Maroon" title="c40">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c41">x1</font>,<font color="Maroon" title="c42">y1</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>and </b><br/><a NAME="E29:22_13"/><i><font color="Green" title="E47">A115</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c31">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c34">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c41">x1</font>
 <b>and </b><br/><a NAME="E30:22_13"/><i><font color="Green" title="E48">A116</font></i>: 
<font color="Maroon" title="c41">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="c31">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c34">g</font>
 ;<br/>
<a NAME="E31:22_13"/><i><font color="Green" title="E49">A117</font></i>: 
<font color="Maroon" title="c40">a2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">x1</font>
 
<b>by </b><i><a class="txt" href="#E28:22_13"><i><font color="Green" title="E46">A114</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E32:22_13"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c31">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c34">g</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c34">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c41">x1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c34">g</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c31">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c41">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="c31">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c31">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c34">g</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E29:22_13"><i><font color="Green" title="E47">A115</font></i></a>, <a class="txt" href="#E30:22_13"><i><font color="Green" title="E48">A116</font></i></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></i>;<br/>
<a NAME="E33:22_13"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c31">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c41">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="c41">x1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c34">g</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c41">x1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c31">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c41">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="c34">g</font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E34:22_13"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c31">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c41">x1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c34">g</font>
 
<b>by </b><i><a class="ref" href="seq_2.html#T9" title="SEQ_2:th.9">SEQ_2:9</a></i>;<br/>
<a NAME="E35:22_13"/><b>then </b>
 <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="c31">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c41">x1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c34">g</font>
 
<b>by </b><i><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></i>;<br/>
<a NAME="E36:22_13"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">x1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c31">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c34">g</font>
 
;<br/>
<a NAME="E37:22_13"/><b>then </b>
<font color="Maroon" title="c40">a2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c31">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </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"><font color="Maroon" title="c31">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </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>
 
<b>by </b><i><a class="txt" href="#E31:22_13"><i><font color="Green" title="E49">A117</font></i></a>, <a class="ref" href="rcomp_1.html#T8" title="RCOMP_1:th.8">RCOMP_1:8</a></i>;<br/>
<b>hence </b><a NAME="E38:22_13"/>
<font color="Maroon" title="c37">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">V</font>
 <b>by </b><i><a class="txt" href="#E9:22_13"><i><font color="Green" title="E40">A108</font></i></a>, <a class="txt" href="#E22:22_13"><i><font color="Green" title="E43">A111</font></i></a>, <a class="txt" href="#E25:22_13"><i><font color="Green" title="E45">A113</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E48:22"/><b>then </b>
<font color="Maroon" title="c20">fX2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c28">fx2</font> = <font color="Maroon" title="c16">fx2</font> as  <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a> <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <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="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#K4" title="BORSUK_1:func.4">:]</a></span> <b>by </b><i><a class="ref" href="topreal6.html#T83" title="TOPREAL6:th.83">TOPREAL6:83</a></i>;<br/>
<a NAME="E50:22"/>
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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</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="c21">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</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="c21">fY2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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"><b>proof </b></a><div class="add">

<b>let </b><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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c21">fY2</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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</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="c21">fY2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c21">fY2</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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</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="c21">fY2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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/>



<b>assume </b><b>that </b><br/><a NAME="E1:22_14"/><i><font color="Green" title="E35">A118</font></i>: 
<font color="Maroon" title="c21">fY2</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>
 <b>and </b><br/><a NAME="E2:22_14"/><i><font color="Green" title="E36">A119</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</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="c21">fY2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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/>

<b>consider </b><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><b> such that </b><br/><a NAME="E4:22_14"/><i><font color="Green" title="E37">A120</font></i>: 
<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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c31">p1</font>,<font color="Maroon" title="c32">p2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>and </b><br/><a NAME="E5:22_14"/><i><font color="Green" title="E38">A121</font></i>: 
<font color="Maroon" title="c21">fY2</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> <font color="Maroon" title="c32">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="#E33:22"><i><font color="Green" title="E34">A28</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c33">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>  <b>by </b><i><a class="txt" href="#E2:22_14"><i><font color="Green" title="E36">A119</font></i></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></i>;<br/>
<b>consider </b><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> <b> such that </b><br/><a NAME="E8:22_14"/><i><font color="Green" title="E39">A122</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c34">g</font>
 <b>and </b><br/><a NAME="E9:22_14"/><i><font color="Green" title="E40">A123</font></i>: 
<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="c32">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </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"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </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="c33">V1</font>
 <b>by </b><i><a class="txt" href="#E1:22_14"><i><font color="Green" title="E35">A118</font></i></a>, <a class="txt" href="#E5:22_14"><i><font color="Green" title="E38">A121</font></i></a>, <a class="ref" href="rcomp_1.html#T40" title="RCOMP_1:th.40">RCOMP_1:40</a></i>;<br/>
<b>reconsider </b><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>  <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c36">W1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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_14"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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_14_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c37">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="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#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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/>

<b>assume </b><a NAME="E1:22_14_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#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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:()"><i><font color="Red">::  thesis: </font></i></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_14_1"/><b>then </b>
 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#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c35">g</font> )
 
;<br/>
<b>hence </b><a NAME="E3:22_14_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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c37">W1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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/>
<b>take </b>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</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">W1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c29">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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">W1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> &amp; <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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">W1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c21">fY2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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">W1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c30">V</font> )</span><br/>

<a NAME="E13:22_14"/><i><font color="Green" title="E41">A124</font></i>: 
<font color="Maroon" title="c32">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/>
<a NAME="E14:22_14"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c35">g</font> )
 
<b>by </b><i><a class="txt" href="#E8:22_14"><i><font color="Green" title="E39">A122</font></i></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></i>;<br/>
<a NAME="E15:22_14"/><b>then </b>
<font color="Maroon" title="c32">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c37">W1</font>
 
<b>by </b><i><a class="txt" href="#E13:22_14"><i><font color="Green" title="E41">A124</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:22_14"/>
<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#K5" title="BORSUK_1:func.5">[:</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">W1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E4:22_14"><i><font color="Green" title="E37">A120</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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">W1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c21">fY2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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">W1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c30">V</font> )</span><br/>

<a NAME="E17:22_14"/>
<font color="Maroon" title="c37">W1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T68" title="PSCOMP_1:th.68">PSCOMP_1:68</a></i>;<br/>
<b>hence </b><a NAME="E18:22_14"/>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</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">W1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T46" title="BORSUK_1:th.46">BORSUK_1:46</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c21">fY2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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">W1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c30">V</font></span><br/>

<b>let </b><font color="Maroon" title="c38">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c38">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">fY2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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">W1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</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="c30">V</font> )</span><br/>

<b>assume </b><a NAME="E19:22_14"/>
<font color="Maroon" title="c38">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">fY2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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">W1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c38">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c30">V</font></span><br/>

<b>then consider </b><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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span><b> such that </b><br/><a NAME="E21:22_14"/><i><font color="Green" title="E42">A125</font></i>: 
<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#K5" title="BORSUK_1:func.5">[:</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">W1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>and </b><br/><a NAME="E22:22_14"/><i><font color="Green" title="E43">A126</font></i>: 
<font color="Maroon" title="c21">fY2</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>
 <b>by </b><i><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<b>consider </b><font color="Maroon" title="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><b> such that </b><br/><a NAME="E24:22_14"/><i><font color="Green" title="E44">A127</font></i>: 
<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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c40">a1</font>,<font color="Maroon" title="c41">a2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>and </b><br/><a NAME="E25:22_14"/><i><font color="Green" title="E45">A128</font></i>: 
<font color="Maroon" title="c21">fY2</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="c41">a2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="#E33:22"><i><font color="Green" title="E34">A28</font></i></a></i>;<br/>
<a NAME="E26:22_14"/>
<font color="Maroon" title="c41">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c37">W1</font>
 
<b>by </b><i><a class="txt" href="#E21:22_14"><i><font color="Green" title="E42">A125</font></i></a>, <a class="txt" href="#E24:22_14"><i><font color="Green" title="E44">A127</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c42">x1</font>, <font color="Maroon" title="c43">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> <b> such that </b><br/><a NAME="E28:22_14"/><i><font color="Green" title="E46">A129</font></i>: 
<font color="Maroon" title="c41">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c42">x1</font>,<font color="Maroon" title="c43">y1</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>and </b><br/><a NAME="E29:22_14"/><i><font color="Green" title="E47">A130</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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">y1</font>
 <b>and </b><br/><a NAME="E30:22_14"/><i><font color="Green" title="E48">A131</font></i>: 
<font color="Maroon" title="c43">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="c32">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="E31:22_14"/><i><font color="Green" title="E49">A132</font></i>: 
<font color="Maroon" title="c41">a2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c43">y1</font>
 
<b>by </b><i><a class="txt" href="#E28:22_14"><i><font color="Green" title="E46">A129</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E32:22_14"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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#K7" title="REAL_1:func.7">+</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">y1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c35">g</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c43">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="c32">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c32">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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> )
 
<b>by </b><i><a class="txt" href="#E29:22_14"><i><font color="Green" title="E47">A130</font></i></a>, <a class="txt" href="#E30:22_14"><i><font color="Green" title="E48">A131</font></i></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></i>;<br/>
<a NAME="E33:22_14"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c43">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">y1</font> <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> <font color="Maroon" title="c43">y1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c43">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="c35">g</font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E34:22_14"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c43">y1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c35">g</font>
 
<b>by </b><i><a class="ref" href="seq_2.html#T9" title="SEQ_2:th.9">SEQ_2:9</a></i>;<br/>
<a NAME="E35:22_14"/><b>then </b>
 <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="c32">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c43">y1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c35">g</font>
 
<b>by </b><i><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></i>;<br/>
<a NAME="E36:22_14"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c43">y1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c35">g</font>
 
;<br/>
<a NAME="E37:22_14"/><b>then </b>
<font color="Maroon" title="c41">a2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c32">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c32">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="txt" href="#E31:22_14"><i><font color="Green" title="E49">A132</font></i></a>, <a class="ref" href="rcomp_1.html#T8" title="RCOMP_1:th.8">RCOMP_1:8</a></i>;<br/>
<b>hence </b><a NAME="E38:22_14"/>
<font color="Maroon" title="c38">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c30">V</font>
 <b>by </b><i><a class="txt" href="#E9:22_14"><i><font color="Green" title="E40">A123</font></i></a>, <a class="txt" href="#E22:22_14"><i><font color="Green" title="E43">A126</font></i></a>, <a class="txt" href="#E25:22_14"><i><font color="Green" title="E45">A128</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E51:22"/><b>then </b>
<font color="Maroon" title="c21">fY2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c29">fy2</font> = <font color="Maroon" title="c17">fy2</font> as  <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a> <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <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="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#K4" title="BORSUK_1:func.4">:]</a></span> <b>by </b><i><a class="ref" href="topreal6.html#T83" title="TOPREAL6:th.83">TOPREAL6:83</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c30">rr</font> = <font color="Maroon" title="c2">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> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c31">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c30">rr</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c32">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c30">rr</font> as  <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a> <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <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="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#K4" title="BORSUK_1:func.4">:]</a></span> <b>by </b><i><a class="txt" href="brouwer.html#E18"><i><font color="Green" title="E13">Lm1</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c33">Zf1</font> = <font color="Maroon" title="c32">f1</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font>;<br/>
<b>set </b><font color="Maroon" title="c34">Zfx2</font> = <font color="Maroon" title="c28">fx2</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font>;<br/>
<b>set </b><font color="Maroon" title="c35">Zfy2</font> = <font color="Maroon" title="c29">fy2</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font>;<br/>
<b>set </b><font color="Maroon" title="c36">Zdx</font> = <font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font>;<br/>
<b>set </b><font color="Maroon" title="c37">Zdy</font> = <font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font>;<br/>
<b>set </b><font color="Maroon" title="c38">Zxo</font> = <font color="Maroon" title="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font>;<br/>
<b>set </b><font color="Maroon" title="c39">Zyo</font> = <font color="Maroon" title="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font>;<br/>
<b>set </b><font color="Maroon" title="c40">xx</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c41">yy</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c42">m</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span></span>)</span>;<br/>
<a NAME="E55:22"/><i><font color="Green" title="E35">A133</font></i>: 
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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">OK</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E56:22"/><i><font color="Green" title="E36">A134</font></i>: 
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="c3">o</font>,<font color="Maroon" title="c2">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">OK</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_15"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c43">y</font>, <font color="Maroon" title="c44">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="c3">o</font>,<font color="Maroon" title="c2">r</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c43">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c44">z</font> implies <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c43">y</font>,<font color="Maroon" title="c44">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">OK</font> )</span><br/>

<b>assume </b><a NAME="E1:22_15"/><i><font color="Green" title="E37">A135</font></i>: 
<font color="Maroon" title="c43">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c44">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c43">y</font>,<font color="Maroon" title="c44">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">OK</font></span><br/>

<a NAME="E2:22_15"/>
( <font color="Maroon" title="c43">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="c44">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> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T55" title="PRE_TOPC:th.55">PRE_TOPC:55</a></i>;<br/>
<a NAME="E3:22_15"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c43">y</font>,<font color="Maroon" title="c44">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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>
 
<b>by </b><i><a class="txt" href="#E1:22_15"><i><font color="Green" title="E37">A135</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:22_15"/>
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c43">y</font>,<font color="Maroon" title="c44">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">OK</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_16"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c43">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c43">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="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c43">b</font> )</span><br/><b>assume </b><a NAME="E1:22_16"/>
<font color="Maroon" title="c43">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="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c43">b</font></span><br/><b>then consider </b><font color="Maroon" title="c44">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:22_16"/><i><font color="Green" title="E37">A136</font></i>: 
<font color="Maroon" title="c44">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:22_16"/><i><font color="Green" title="E38">A137</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c43">b</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:22_16"/>
<font color="Maroon" title="c44">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>
 <b>by </b><i><a class="txt" href="#E55:22"><i><font color="Green" title="E35">A133</font></i></a>, <a class="txt" href="#E3:22_16"><i><font color="Green" title="E37">A136</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon" title="c45">y</font>, <font color="Maroon" title="c46">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><b> such that </b><br/><a NAME="E7:22_16"/><i><font color="Green" title="E39">A138</font></i>: 
<font color="Maroon" title="c44">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c45">y</font>,<font color="Maroon" title="c46">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>and </b><br/><a NAME="E8:22_16"/><i><font color="Green" title="E40">A139</font></i>: 
<font color="Maroon" title="c45">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c46">z</font>
 ;<br/><a NAME="E9:22_16"/>
<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"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</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="c3">o</font>,<font color="Maroon" title="c2">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="c3">o</font>,<font color="Maroon" title="c2">r</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <b>by </b><i><a class="txt" href="#E55:22"><i><font color="Green" title="E35">A133</font></i></a>, <a class="txt" href="#E3:22_16"><i><font color="Green" title="E37">A136</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon" title="c47">a1</font>, <font color="Maroon" title="c48">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="c3">o</font>,<font color="Maroon" title="c2">r</font></span>)</span><b> such that </b><br/><a NAME="E11:22_16"/><i><font color="Green" title="E41">A140</font></i>: 
<font color="Maroon" title="c44">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c47">a1</font>,<font color="Maroon" title="c48">a2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50" title="BORSUK_1:th.50">BORSUK_1:50</a></i>;<br/><a NAME="E12:22_16"/><i><font color="Green" title="E42">A141</font></i>: 
( <font color="Maroon" title="c47">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c45">y</font> &amp; <font color="Maroon" title="c48">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c46">z</font> )
 <b>by </b><i><a class="txt" href="#E7:22_16"><i><font color="Green" title="E39">A138</font></i></a>, <a class="txt" href="#E11:22_16"><i><font color="Green" title="E41">A140</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>consider </b><font color="Maroon" title="c49">x3</font>, <font color="Maroon" title="c50">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><b> such that </b><br/><a NAME="E14:22_16"/><i><font color="Green" title="E43">A142</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c45">y</font>,<font color="Maroon" title="c46">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c49">x3</font>,<font color="Maroon" title="c50">x4</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>and </b><br/><a NAME="E15:22_16"/><i><font color="Green" title="E44">A143</font></i>: 
<font color="Maroon" title="c26">dx</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c45">y</font>,<font color="Maroon" title="c46">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">x3</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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">x4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E24:22"><i><font color="Green" title="E28">A19</font></i></a></i>;<br/><a NAME="E16:22_16"/><i><font color="Green" title="E45">A144</font></i>: 
( <font color="Maroon" title="c45">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c49">x3</font> &amp; <font color="Maroon" title="c46">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c50">x4</font> )
 <b>by </b><i><a class="txt" href="#E14:22_16"><i><font color="Green" title="E43">A142</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>consider </b><font color="Maroon" title="c51">x5</font>, <font color="Maroon" title="c52">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><b> such that </b><br/><a NAME="E18:22_16"/><i><font color="Green" title="E46">A145</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c45">y</font>,<font color="Maroon" title="c46">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c51">x5</font>,<font color="Maroon" title="c52">x6</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>and </b><br/><a NAME="E19:22_16"/><i><font color="Green" title="E47">A146</font></i>: 
<font color="Maroon" title="c27">dy</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c45">y</font>,<font color="Maroon" title="c46">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c51">x5</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c52">x6</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E27:22"><i><font color="Green" title="E30">A22</font></i></a></i>;<br/><a NAME="E20:22_16"/><i><font color="Green" title="E48">A147</font></i>: 
( <font color="Maroon" title="c45">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c51">x5</font> &amp; <font color="Maroon" title="c46">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c52">x6</font> )
 <b>by </b><i><a class="txt" href="#E18:22_16"><i><font color="Green" title="E46">A145</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>set </b><font color="Maroon" title="c53">r1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c45">y</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c46">z</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>;<br/><b>set </b><font color="Maroon" title="c54">r2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c45">y</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c46">z</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>;<br/><a NAME="E21:22_16"/><i><font color="Green" title="E49">A148</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c45">y</font>,<font color="Maroon" title="c46">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">dx</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c45">y</font>,<font color="Maroon" title="c46">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>by </b><i><a class="txt" href="#E56:22"><i><font color="Green" title="E36">A134</font></i></a>, <a class="txt" href="#E8:22_16"><i><font color="Green" title="E40">A139</font></i></a>, <a class="txt" href="#E12:22_16"><i><font color="Green" title="E42">A141</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><a NAME="E22:22_16"/><i><font color="Green" title="E50">A149</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c45">y</font>,<font color="Maroon" title="c46">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">dy</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c45">y</font>,<font color="Maroon" title="c46">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>by </b><i><a class="txt" href="#E56:22"><i><font color="Green" title="E36">A134</font></i></a>, <a class="txt" href="#E8:22_16"><i><font color="Green" title="E40">A139</font></i></a>, <a class="txt" href="#E12:22_16"><i><font color="Green" title="E42">A141</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><i><font color="Green" title="E51">A150</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c45">y</font>,<font color="Maroon" title="c46">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c45">y</font>,<font color="Maroon" title="c46">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c45">y</font>,<font color="Maroon" title="c46">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="#E3:22_16"><i><font color="Green" title="E37">A136</font></i></a>, <a class="txt" href="#E7:22_16"><i><font color="Green" title="E39">A138</font></i></a>, <a class="ref" href="valued_1.html#T1" title="VALUED_1:th.1">VALUED_1:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c45">y</font>,<font color="Maroon" title="c46">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c45">y</font>,<font color="Maroon" title="c46">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c45">y</font>,<font color="Maroon" title="c46">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c45">y</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c46">z</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c45">y</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c46">z</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E15:22_16"><i><font color="Green" title="E44">A143</font></i></a>, <a class="txt" href="#E16:22_16"><i><font color="Green" title="E45">A144</font></i></a>, <a class="txt" href="#E19:22_16"><i><font color="Green" title="E47">A146</font></i></a>, <a class="txt" href="#E20:22_16"><i><font color="Green" title="E48">A147</font></i></a>, <a class="txt" href="#E21:22_16"><i><font color="Green" title="E49">A148</font></i></a>, <a class="txt" href="#E22:22_16"><i><font color="Green" title="E50">A149</font></i></a>, <a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_16_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:22_16_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c45">y</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c46">z</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c45">y</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c46">z</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:22_16_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c45">y</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c46">z</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c45">y</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c46">z</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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>  )
 <b>by </b><i><a class="ref" href="complex1.html#T2" title="COMPLEX1:th.2">COMPLEX1:2</a></i>;<br/><b>hence </b><a NAME="E3:22_16_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E8:22_16"><i><font color="Green" title="E40">A139</font></i></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E25:22_16"/>
 <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="c43">b</font>
 <b>by </b><i><a class="txt" href="#E4:22_16"><i><font color="Green" title="E38">A137</font></i></a>, <a class="txt" href="#E7:22_16"><i><font color="Green" title="E39">A138</font></i></a>, <a class="txt" href="#E23:22_16"><i><font color="Green" title="E51">A150</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c43">m</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span></span>)</span> as  <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <b>by </b><i><a class="ref" href="partfun3.html#D1" title="PARTFUN3:def.1">PARTFUN3:def 1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c44">p1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c45">p2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c32">f1</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span>;<br/>
<a NAME="E59:22"/><i><font color="Green" title="E37">A152</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c32">f1</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">OK</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_17"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c32">f1</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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/><b>assume </b><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"><span class="p4">(<span class="default"><font color="Maroon" title="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c32">f1</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/><b>then consider </b><font color="Maroon" title="c47">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:22_17"/><i><font color="Green" title="E38">A153</font></i>: 
<font color="Maroon" title="c47">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c32">f1</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:22_17"/><i><font color="Green" title="E39">A154</font></i>: 
<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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c32">f1</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<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>
 <b>by </b><i><a class="txt" href="#E55:22"><i><font color="Green" title="E35">A133</font></i></a>, <a class="txt" href="#E3:22_17"><i><font color="Green" title="E38">A153</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><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><b> such that </b><br/><a NAME="E7:22_17"/><i><font color="Green" title="E40">A155</font></i>: 
<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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>and </b><br/><a NAME="E8:22_17"/><i><font color="Green" title="E41">A156</font></i>: 
<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/><b>set </b><font color="Maroon" title="c50">r3</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c49">z</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>;<br/><b>set </b><font color="Maroon" title="c51">r4</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c49">z</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>;<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#K4" title="BORSUK_1:func.4">[:</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="c3">o</font>,<font color="Maroon" title="c2">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="c3">o</font>,<font color="Maroon" title="c2">r</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <b>by </b><i><a class="txt" href="#E55:22"><i><font color="Green" title="E35">A133</font></i></a>, <a class="txt" href="#E3:22_17"><i><font color="Green" title="E38">A153</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon" title="c52">a1</font>, <font color="Maroon" title="c53">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="c3">o</font>,<font color="Maroon" title="c2">r</font></span>)</span><b> such that </b><br/><a NAME="E11:22_17"/><i><font color="Green" title="E42">A157</font></i>: 
<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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c52">a1</font>,<font color="Maroon" title="c53">a2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50" title="BORSUK_1:th.50">BORSUK_1:50</a></i>;<br/><a NAME="E12:22_17"/><i><font color="Green" title="E43">A158</font></i>: 
( <font color="Maroon" title="c52">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c48">y</font> &amp; <font color="Maroon" title="c53">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c49">z</font> )
 <b>by </b><i><a class="txt" href="#E7:22_17"><i><font color="Green" title="E40">A155</font></i></a>, <a class="txt" href="#E11:22_17"><i><font color="Green" title="E42">A157</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E13:22_17"/><b>then </b><i><font color="Green" title="E44">A159</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c32">f1</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>by </b><i><a class="txt" href="#E56:22"><i><font color="Green" title="E36">A134</font></i></a>, <a class="txt" href="#E8:22_17"><i><font color="Green" title="E41">A156</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><a NAME="E14:22_17"/><i><font color="Green" title="E45">A160</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">xo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>by </b><i><a class="txt" href="#E56:22"><i><font color="Green" title="E36">A134</font></i></a>, <a class="txt" href="#E8:22_17"><i><font color="Green" title="E41">A156</font></i></a>, <a class="txt" href="#E12:22_17"><i><font color="Green" title="E43">A158</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><a NAME="E15:22_17"/><i><font color="Green" title="E46">A161</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">yo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>by </b><i><a class="txt" href="#E56:22"><i><font color="Green" title="E36">A134</font></i></a>, <a class="txt" href="#E8:22_17"><i><font color="Green" title="E41">A156</font></i></a>, <a class="txt" href="#E12:22_17"><i><font color="Green" title="E43">A158</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><b>consider </b><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><b> such that </b><br/><a NAME="E17:22_17"/><i><font color="Green" title="E47">A162</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c54">x9</font>,<font color="Maroon" title="c55">x10</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>and </b><br/><a NAME="E18:22_17"/><i><font color="Green" title="E48">A163</font></i>: 
<font color="Maroon" title="c24">xo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E18:22"><i><font color="Green" title="E24">A13</font></i></a></i>;<br/><a NAME="E19:22_17"/><i><font color="Green" title="E49">A164</font></i>: 
( <font color="Maroon" title="c48">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c54">x9</font> &amp; <font color="Maroon" title="c49">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c55">x10</font> )
 <b>by </b><i><a class="txt" href="#E17:22_17"><i><font color="Green" title="E47">A162</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>consider </b><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><b> such that </b><br/><a NAME="E21:22_17"/><i><font color="Green" title="E50">A165</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">x11</font>,<font color="Maroon" title="c57">x12</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>and </b><br/><a NAME="E22:22_17"/><i><font color="Green" title="E51">A166</font></i>: 
<font color="Maroon" title="c25">yo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E21:22"><i><font color="Green" title="E26">A16</font></i></a></i>;<br/><a NAME="E23:22_17"/><i><font color="Green" title="E52">A167</font></i>: 
( <font color="Maroon" title="c48">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c56">x11</font> &amp; <font color="Maroon" title="c49">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c57">x12</font> )
 <b>by </b><i><a class="txt" href="#E21:22_17"><i><font color="Green" title="E50">A165</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><i><font color="Green" title="E53">A168</font></i>: <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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c32">f1</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c32">f1</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="#E3:22_17"><i><font color="Green" title="E38">A153</font></i></a>, <a class="txt" href="#E7:22_17"><i><font color="Green" title="E40">A155</font></i></a>, <a class="ref" href="valued_1.html#T13" title="VALUED_1:th.13">VALUED_1:13</a></i>
<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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c2">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E13:22_17"><i><font color="Green" title="E44">A159</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c2">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E3:22_17"><i><font color="Green" title="E38">A153</font></i></a>, <a class="txt" href="#E7:22_17"><i><font color="Green" title="E40">A155</font></i></a>, <a class="ref" href="valued_1.html#T1" title="VALUED_1:th.1">VALUED_1:1</a></i>
<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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c48">y</font>,<font color="Maroon" title="c49">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c2">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></i>
<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#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c2">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E14:22_17"><i><font color="Green" title="E45">A160</font></i></a>, <a class="txt" href="#E15:22_17"><i><font color="Green" title="E46">A161</font></i></a>, <a class="txt" href="#E18:22_17"><i><font color="Green" title="E48">A163</font></i></a>, <a class="txt" href="#E19:22_17"><i><font color="Green" title="E49">A164</font></i></a>, <a class="txt" href="#E22:22_17"><i><font color="Green" title="E51">A166</font></i></a>, <a class="txt" href="#E23:22_17"><i><font color="Green" title="E52">A167</font></i></a>, <a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></i>
;<br/><a NAME="E25:22_17"/>
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="c3">o</font>,<font color="Maroon" title="c2">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="c3">o</font>,<font color="Maroon" title="c2">r</font>
 <b>by </b><i><a class="ref" href="brouwer.html#T3" target="_self" title="BROUWER:th.3">Th3</a></i>;<br/><a NAME="E26:22_17"/><b>then </b>
<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="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c3">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="c2">r</font>
 <b>by </b><i><a class="txt" href="#E12:22_17"><i><font color="Green" title="E43">A158</font></i></a>, <a class="ref" href="topreal9.html#T8" title="TOPREAL9:th.8">TOPREAL9:8</a></i>;<br/><a NAME="E27:22_17"/><b>then </b><i><font color="Green" title="E54">A169</font></i>: 
<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="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c3">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="c2">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> 
 <b>by </b><i><a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></i>;<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="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c3">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="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c3">o</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`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="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c3">o</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="jgraph_1.html#T46" title="JGRAPH_1:th.46">JGRAPH_1:46</a></i>
<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#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c3">o</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T8" title="TOPREAL3:th.8">TOPREAL3:8</a></i>
<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#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T8" title="TOPREAL3:th.8">TOPREAL3:8</a></i>
;<br/><a NAME="E29:22_17"/><b>then </b>
<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#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c2">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="c2">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="c2">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E27:22_17"><i><font color="Green" title="E54">A169</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>hence </b><a NAME="E30:22_17"/>
 <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>
 <b>by </b><i><a class="txt" href="#E4:22_17"><i><font color="Green" title="E39">A154</font></i></a>, <a class="txt" href="#E7:22_17"><i><font color="Green" title="E40">A155</font></i></a>, <a class="txt" href="#E24:22_17"><i><font color="Green" title="E53">A168</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c46">p2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c32">f1</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> as  <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <b>by </b><i><a class="ref" href="partfun3.html#D3" title="PARTFUN3:def.3">PARTFUN3:def 3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c47">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c43">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c46">p2</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c48">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c43">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c46">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14" title="PARTFUN3:func.14">/</a> <font color="Maroon" title="c43">m</font>;<br/>
<b>set </b><font color="Maroon" title="c49">x3</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c28">fx2</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c43">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c46">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14" title="PARTFUN3:func.14">/</a> <font color="Maroon" title="c43">m</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c50">y3</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c29">fy2</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c43">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c46">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14" title="PARTFUN3:func.14">/</a> <font color="Maroon" title="c43">m</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c51">X3</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c28">fx2</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c43">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c46">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14" title="PARTFUN3:func.14">/</a> <font color="Maroon" title="c43">m</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span></span>)</span>, <font color="Maroon" title="c52">Y3</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c29">fy2</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c43">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c46">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14" title="PARTFUN3:func.14">/</a> <font color="Maroon" title="c43">m</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">OK</font></span>)</span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<b>set </b><font color="Maroon" title="c53">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="c51">X3</font>,<font color="Maroon" title="c52">Y3</font></span><a href="funct_3.html#K14" title="FUNCT_3:func.14">:&gt;</a></span>;<br/>
<b>set </b><font color="Maroon" title="c54">R</font> =  <a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a> ;<br/>
<a NAME="E63:22"/><i><font color="Green" title="E38">A170</font></i>: 
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="c3">o</font>,<font color="Maroon" title="c2">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="c10">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="c51">X3</font>,<font color="Maroon" title="c52">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Olive" title="b1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">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#K6" title="BORSUK_1:func.6">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_18"><b>proof </b></a><div class="add">

<b>let </b><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="c3">o</font>,<font color="Maroon" title="c2">r</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c10">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="c51">X3</font>,<font color="Maroon" title="c52">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c55">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">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#K6" title="BORSUK_1:func.6">]</a></span></span><br/>

<a NAME="E1:22_18"/><i><font color="Green" title="E39">A171</font></i>: 
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="c10">f</font>
 
<b>by </b><i><a class="txt" href="#E15:22"><i><font color="Green" title="E22">A10</font></i></a>, <a class="ref" href="abian.html#D5" title="ABIAN:def.5">ABIAN:def 5</a></i>;<br/>
<b>then consider </b><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><b> such that </b><br/><a NAME="E3:22_18"/><i><font color="Green" title="E40">A172</font></i>: 
<font color="Maroon" title="c56">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c55">x</font>
 <b>and </b><br/><a NAME="E4:22_18"/><i><font color="Green" title="E41">A173</font></i>: 
<font color="Maroon" title="c57">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c55">x</font>
 <b>and </b><br/><a NAME="E5:22_18"/><i><font color="Green" title="E42">A174</font></i>: 
 <a href="brouwer.html#K4" title="BROUWER:func.4">HC</a> <font color="Maroon" title="c55">x</font>,<font color="Maroon" title="c10">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="c3">o</font>,<font color="Maroon" title="c2">r</font>
 <b>by </b><i><a class="ref" href="brouwer.html#D4" target="_self" title="BROUWER:def.4">Def4</a></i>;<br/>
<a NAME="E6:22_18"/><i><font color="Green" title="E43">A175</font></i>: 
<font color="Maroon" title="c55">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c55">x</font>
 
<b>by </b><i><a class="txt" href="#E1:22_18"><i><font color="Green" title="E39">A171</font></i></a>, <a class="ref" href="abian.html#D4" title="ABIAN:def.4">ABIAN:def 4</a></i>;<br/>
<a NAME="E7:22_18"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c55">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">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#K6" title="BORSUK_1:func.6">]</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>
 
<b>by </b><i><a class="txt" href="#E3:22_18"><i><font color="Green" title="E40">A172</font></i></a>, <a class="txt" href="#E4:22_18"><i><font color="Green" title="E41">A173</font></i></a></i>;<br/>
<a NAME="E8:22_18"/><b>then </b><i><font color="Green" title="E44">A176</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">OK</font>
 
<b>by </b><i><a class="txt" href="#E3:22_18"><i><font color="Green" title="E40">A172</font></i></a>, <a class="txt" href="#E4:22_18"><i><font color="Green" title="E41">A173</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>set </b><font color="Maroon" title="c58">r1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`1</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c59">r2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c56">y</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`2</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c60">r3</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c61">r4</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c62">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#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c2">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#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c63">x1</font>, <font color="Maroon" title="c64">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><b> such that </b><br/><a NAME="E10:22_18"/><i><font color="Green" title="E45">A177</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c63">x1</font>,<font color="Maroon" title="c64">x2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>and </b><br/><a NAME="E11:22_18"/><i><font color="Green" title="E46">A178</font></i>: 
<font color="Maroon" title="c28">fx2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c64">x2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="#E30:22"><i><font color="Green" title="E32">A25</font></i></a></i>;<br/>
<a NAME="E12:22_18"/><i><font color="Green" title="E47">A179</font></i>: 
<font color="Maroon" title="c57">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c64">x2</font>
 
<b>by </b><i><a class="txt" href="#E10:22_18"><i><font color="Green" title="E45">A177</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c65">x7</font>, <font color="Maroon" title="c66">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><b> such that </b><br/><a NAME="E14:22_18"/><i><font color="Green" title="E48">A180</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c65">x7</font>,<font color="Maroon" title="c66">x8</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>and </b><br/><a NAME="E15:22_18"/><i><font color="Green" title="E49">A181</font></i>: 
<font color="Maroon" title="c29">fy2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c66">x8</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="#E33:22"><i><font color="Green" title="E34">A28</font></i></a></i>;<br/>
<a NAME="E16:22_18"/><i><font color="Green" title="E50">A182</font></i>: 
<font color="Maroon" title="c57">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c66">x8</font>
 
<b>by </b><i><a class="txt" href="#E14:22_18"><i><font color="Green" title="E48">A180</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c67">x3</font>, <font color="Maroon" title="c68">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><b> such that </b><br/><a NAME="E18:22_18"/><i><font color="Green" title="E51">A183</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c67">x3</font>,<font color="Maroon" title="c68">x4</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>and </b><br/><a NAME="E19:22_18"/><i><font color="Green" title="E52">A184</font></i>: 
<font color="Maroon" title="c26">dx</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c67">x3</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c68">x4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E24:22"><i><font color="Green" title="E28">A19</font></i></a></i>;<br/>
<a NAME="E20:22_18"/><i><font color="Green" title="E53">A185</font></i>: 
( <font color="Maroon" title="c56">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c67">x3</font> &amp; <font color="Maroon" title="c57">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c68">x4</font> )
 
<b>by </b><i><a class="txt" href="#E18:22_18"><i><font color="Green" title="E51">A183</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c69">x5</font>, <font color="Maroon" title="c70">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><b> such that </b><br/><a NAME="E22:22_18"/><i><font color="Green" title="E54">A186</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c69">x5</font>,<font color="Maroon" title="c70">x6</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>and </b><br/><a NAME="E23:22_18"/><i><font color="Green" title="E55">A187</font></i>: 
<font color="Maroon" title="c27">dy</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c69">x5</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c70">x6</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E27:22"><i><font color="Green" title="E30">A22</font></i></a></i>;<br/>
<a NAME="E24:22_18"/><i><font color="Green" title="E56">A188</font></i>: 
( <font color="Maroon" title="c56">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c69">x5</font> &amp; <font color="Maroon" title="c57">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c70">x6</font> )
 
<b>by </b><i><a class="txt" href="#E22:22_18"><i><font color="Green" title="E54">A186</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c71">x9</font>, <font color="Maroon" title="c72">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><b> such that </b><br/><a NAME="E26:22_18"/><i><font color="Green" title="E57">A189</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c71">x9</font>,<font color="Maroon" title="c72">x10</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>and </b><br/><a NAME="E27:22_18"/><i><font color="Green" title="E58">A190</font></i>: 
<font color="Maroon" title="c24">xo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c72">x10</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E18:22"><i><font color="Green" title="E24">A13</font></i></a></i>;<br/>
<a NAME="E28:22_18"/><i><font color="Green" title="E59">A191</font></i>: 
( <font color="Maroon" title="c56">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c71">x9</font> &amp; <font color="Maroon" title="c57">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c72">x10</font> )
 
<b>by </b><i><a class="txt" href="#E26:22_18"><i><font color="Green" title="E57">A189</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c73">x11</font>, <font color="Maroon" title="c74">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><b> such that </b><br/><a NAME="E30:22_18"/><i><font color="Green" title="E60">A192</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c73">x11</font>,<font color="Maroon" title="c74">x12</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>and </b><br/><a NAME="E31:22_18"/><i><font color="Green" title="E61">A193</font></i>: 
<font color="Maroon" title="c25">yo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c74">x12</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E21:22"><i><font color="Green" title="E26">A16</font></i></a></i>;<br/>
<a NAME="E32:22_18"/><i><font color="Green" title="E62">A194</font></i>: 
( <font color="Maroon" title="c56">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c73">x11</font> &amp; <font color="Maroon" title="c57">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c74">x12</font> )
 
<b>by </b><i><a class="txt" href="#E30:22_18"><i><font color="Green" title="E60">A192</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E33:22_18"/><i><font color="Green" title="E63">A195</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c51">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">OK</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E34:22_18"/><i><font color="Green" title="E64">A196</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c52">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">OK</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E35:22_18"/><i><font color="Green" title="E65">A197</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c43">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c46">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">OK</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E36:22_18"/><i><font color="Green" title="E66">A198</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">dx</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 
<b>by </b><i><a class="txt" href="#E56:22"><i><font color="Green" title="E36">A134</font></i></a>, <a class="txt" href="#E3:22_18"><i><font color="Green" title="E40">A172</font></i></a>, <a class="txt" href="#E4:22_18"><i><font color="Green" title="E41">A173</font></i></a>, <a class="txt" href="#E6:22_18"><i><font color="Green" title="E43">A175</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<a NAME="E37:22_18"/><i><font color="Green" title="E67">A199</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">dy</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 
<b>by </b><i><a class="txt" href="#E56:22"><i><font color="Green" title="E36">A134</font></i></a>, <a class="txt" href="#E3:22_18"><i><font color="Green" title="E40">A172</font></i></a>, <a class="txt" href="#E4:22_18"><i><font color="Green" title="E41">A173</font></i></a>, <a class="txt" href="#E6:22_18"><i><font color="Green" title="E43">A175</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<a NAME="E38:22_18"/><i><font color="Green" title="E68">A200</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c32">f1</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 
<b>by </b><i><a class="txt" href="#E56:22"><i><font color="Green" title="E36">A134</font></i></a>, <a class="txt" href="#E3:22_18"><i><font color="Green" title="E40">A172</font></i></a>, <a class="txt" href="#E4:22_18"><i><font color="Green" title="E41">A173</font></i></a>, <a class="txt" href="#E6:22_18"><i><font color="Green" title="E43">A175</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<a NAME="E39:22_18"/><i><font color="Green" title="E69">A201</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">xo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 
<b>by </b><i><a class="txt" href="#E56:22"><i><font color="Green" title="E36">A134</font></i></a>, <a class="txt" href="#E3:22_18"><i><font color="Green" title="E40">A172</font></i></a>, <a class="txt" href="#E4:22_18"><i><font color="Green" title="E41">A173</font></i></a>, <a class="txt" href="#E6:22_18"><i><font color="Green" title="E43">A175</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<a NAME="E40:22_18"/><i><font color="Green" title="E70">A202</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">yo</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 
<b>by </b><i><a class="txt" href="#E56:22"><i><font color="Green" title="E36">A134</font></i></a>, <a class="txt" href="#E3:22_18"><i><font color="Green" title="E40">A172</font></i></a>, <a class="txt" href="#E4:22_18"><i><font color="Green" title="E41">A173</font></i></a>, <a class="txt" href="#E6:22_18"><i><font color="Green" title="E43">A175</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<i><font color="Green" title="E71">A203</font></i>: <font color="Maroon" title="c43">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="#E55:22"><i><font color="Green" title="E35">A133</font></i></a>, <a class="txt" href="#E8:22_18"><i><font color="Green" title="E44">A176</font></i></a>, <a class="ref" href="valued_1.html#T1" title="VALUED_1:th.1">VALUED_1:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></i>
<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#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E19:22_18"><i><font color="Green" title="E52">A184</font></i></a>, <a class="txt" href="#E20:22_18"><i><font color="Green" title="E53">A185</font></i></a>, <a class="txt" href="#E23:22_18"><i><font color="Green" title="E55">A187</font></i></a>, <a class="txt" href="#E24:22_18"><i><font color="Green" title="E56">A188</font></i></a>, <a class="txt" href="#E36:22_18"><i><font color="Green" title="E66">A198</font></i></a>, <a class="txt" href="#E37:22_18"><i><font color="Green" title="E67">A199</font></i></a>, <a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></i>
;<br/>
<a NAME="E42:22_18"/><i><font color="Green" title="E72">A204</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></i>;<br/>
<a NAME="E43:22_18"/><i><font color="Green" title="E73">A205</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></i>;<br/>
<a NAME="E44:22_18"/><i><font color="Green" title="E74">A206</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E55:22"><i><font color="Green" title="E35">A133</font></i></a>, <a class="txt" href="#E8:22_18"><i><font color="Green" title="E44">A176</font></i></a>, <a class="ref" href="valued_1.html#T1" title="VALUED_1:th.1">VALUED_1:1</a></i>;<br/>
<a NAME="E45:22_18"/><b>then </b><i><font color="Green" title="E75">A207</font></i>: 
<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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="#E19:22_18"><i><font color="Green" title="E52">A184</font></i></a>, <a class="txt" href="#E20:22_18"><i><font color="Green" title="E53">A185</font></i></a>, <a class="txt" href="#E23:22_18"><i><font color="Green" title="E55">A187</font></i></a>, <a class="txt" href="#E24:22_18"><i><font color="Green" title="E56">A188</font></i></a>, <a class="txt" href="#E27:22_18"><i><font color="Green" title="E58">A190</font></i></a>, <a class="txt" href="#E28:22_18"><i><font color="Green" title="E59">A191</font></i></a>, <a class="txt" href="#E31:22_18"><i><font color="Green" title="E61">A193</font></i></a>, <a class="txt" href="#E32:22_18"><i><font color="Green" title="E62">A194</font></i></a>, <a class="txt" href="#E36:22_18"><i><font color="Green" title="E66">A198</font></i></a>, <a class="txt" href="#E37:22_18"><i><font color="Green" title="E67">A199</font></i></a>, <a class="txt" href="#E39:22_18"><i><font color="Green" title="E69">A201</font></i></a>, <a class="txt" href="#E40:22_18"><i><font color="Green" title="E70">A202</font></i></a>, <a class="txt" href="#E42:22_18"><i><font color="Green" title="E72">A204</font></i></a>, <a class="txt" href="#E43:22_18"><i><font color="Green" title="E73">A205</font></i></a>, <a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></i>;<br/>
<i><font color="Green" title="E76">A208</font></i>: <font color="Maroon" title="c46">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c32">f1</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="#E55:22"><i><font color="Green" title="E35">A133</font></i></a>, <a class="txt" href="#E59:22"><i><font color="Green" title="E37">A152</font></i></a>, <a class="txt" href="#E8:22_18"><i><font color="Green" title="E44">A176</font></i></a>, <a class="ref" href="valued_1.html#T13" title="VALUED_1:th.13">VALUED_1:13</a></i>
<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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c2">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E38:22_18"><i><font color="Green" title="E68">A200</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c2">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E55:22"><i><font color="Green" title="E35">A133</font></i></a>, <a class="txt" href="#E8:22_18"><i><font color="Green" title="E44">A176</font></i></a>, <a class="ref" href="valued_1.html#T1" title="VALUED_1:th.1">VALUED_1:1</a></i>
<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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c2">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></i>
<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#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c2">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E27:22_18"><i><font color="Green" title="E58">A190</font></i></a>, <a class="txt" href="#E28:22_18"><i><font color="Green" title="E59">A191</font></i></a>, <a class="txt" href="#E31:22_18"><i><font color="Green" title="E61">A193</font></i></a>, <a class="txt" href="#E32:22_18"><i><font color="Green" title="E62">A194</font></i></a>, <a class="txt" href="#E39:22_18"><i><font color="Green" title="E69">A201</font></i></a>, <a class="txt" href="#E40:22_18"><i><font color="Green" title="E70">A202</font></i></a>, <a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></i>
;<br/>
<a NAME="E47:22_18"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c43">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c46">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">OK</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><i><font color="Green" title="E77">A209</font></i>: <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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c43">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c46">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c43">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c46">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="#E55:22"><i><font color="Green" title="E35">A133</font></i></a>, <a class="txt" href="#E8:22_18"><i><font color="Green" title="E44">A176</font></i></a>, <a class="ref" href="partfun3.html#D5" title="PARTFUN3:def.5">PARTFUN3:def 5</a></i>
<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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c43">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c46">p2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E55:22"><i><font color="Green" title="E35">A133</font></i></a>, <a class="txt" href="#E8:22_18"><i><font color="Green" title="E44">A176</font></i></a>, <a class="txt" href="#E35:22_18"><i><font color="Green" title="E65">A197</font></i></a>, <a class="ref" href="valued_1.html#T13" title="VALUED_1:th.13">VALUED_1:13</a></i>
<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#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c2">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E41:22_18"><i><font color="Green" title="E71">A203</font></i></a>, <a class="txt" href="#E45:22_18"><i><font color="Green" title="E75">A207</font></i></a>, <a class="txt" href="#E46:22_18"><i><font color="Green" title="E76">A208</font></i></a>, <a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></i>
;<br/>
<a NAME="E49:22_18"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c43">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c46">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14" title="PARTFUN3:func.14">/</a> <font color="Maroon" title="c43">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">OK</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><i><font color="Green" title="E78">A210</font></i>: <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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c43">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c46">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14" title="PARTFUN3:func.14">/</a> <font color="Maroon" title="c43">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c43">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c46">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c43">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E55:22"><i><font color="Green" title="E35">A133</font></i></a>, <a class="txt" href="#E8:22_18"><i><font color="Green" title="E44">A176</font></i></a>, <a class="ref" href="rfunct_1.html#D4" title="RFUNCT_1:def.4">RFUNCT_1:def 4</a></i>
<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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c43">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c46">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c43">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></i>
<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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c43">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c46">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E55:22"><i><font color="Green" title="E35">A133</font></i></a>, <a class="txt" href="#E8:22_18"><i><font color="Green" title="E44">A176</font></i></a>, <a class="txt" href="#E41:22_18"><i><font color="Green" title="E71">A203</font></i></a>, <a class="ref" href="valued_1.html#T1" title="VALUED_1:th.1">VALUED_1:1</a></i>
<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#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c2">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#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E19:22_18"><i><font color="Green" title="E52">A184</font></i></a>, <a class="txt" href="#E20:22_18"><i><font color="Green" title="E53">A185</font></i></a>, <a class="txt" href="#E23:22_18"><i><font color="Green" title="E55">A187</font></i></a>, <a class="txt" href="#E24:22_18"><i><font color="Green" title="E56">A188</font></i></a>, <a class="txt" href="#E27:22_18"><i><font color="Green" title="E58">A190</font></i></a>, <a class="txt" href="#E28:22_18"><i><font color="Green" title="E59">A191</font></i></a>, <a class="txt" href="#E31:22_18"><i><font color="Green" title="E61">A193</font></i></a>, <a class="txt" href="#E32:22_18"><i><font color="Green" title="E62">A194</font></i></a>, <a class="txt" href="#E36:22_18"><i><font color="Green" title="E66">A198</font></i></a>, <a class="txt" href="#E37:22_18"><i><font color="Green" title="E67">A199</font></i></a>, <a class="txt" href="#E39:22_18"><i><font color="Green" title="E69">A201</font></i></a>, <a class="txt" href="#E40:22_18"><i><font color="Green" title="E70">A202</font></i></a>, <a class="txt" href="#E42:22_18"><i><font color="Green" title="E72">A204</font></i></a>, <a class="txt" href="#E43:22_18"><i><font color="Green" title="E73">A205</font></i></a>, <a class="txt" href="#E44:22_18"><i><font color="Green" title="E74">A206</font></i></a>, <a class="txt" href="#E48:22_18"><i><font color="Green" title="E77">A209</font></i></a>, <a class="ref" href="valued_1.html#T8" title="VALUED_1:th.8">VALUED_1:8</a></i>
;<br/>
<i><font color="Green" title="E79">A211</font></i>: <font color="Maroon" title="c51">X3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">fx2</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c43">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c46">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14" title="PARTFUN3:func.14">/</a> <font color="Maroon" title="c43">m</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="#E55:22"><i><font color="Green" title="E35">A133</font></i></a>, <a class="txt" href="#E8:22_18"><i><font color="Green" title="E44">A176</font></i></a>, <a class="ref" href="valued_1.html#T1" title="VALUED_1:th.1">VALUED_1:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c43">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c46">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14" title="PARTFUN3:func.14">/</a> <font color="Maroon" title="c43">m</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="#E56:22"><i><font color="Green" title="E36">A134</font></i></a>, <a class="txt" href="#E3:22_18"><i><font color="Green" title="E40">A172</font></i></a>, <a class="txt" href="#E4:22_18"><i><font color="Green" title="E41">A173</font></i></a>, <a class="txt" href="#E6:22_18"><i><font color="Green" title="E43">A175</font></i></a>, <a class="txt" href="#E11:22_18"><i><font color="Green" title="E46">A178</font></i></a>, <a class="txt" href="#E12:22_18"><i><font color="Green" title="E47">A179</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c2">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#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E19:22_18"><i><font color="Green" title="E52">A184</font></i></a>, <a class="txt" href="#E20:22_18"><i><font color="Green" title="E53">A185</font></i></a>, <a class="txt" href="#E36:22_18"><i><font color="Green" title="E66">A198</font></i></a>, <a class="txt" href="#E50:22_18"><i><font color="Green" title="E78">A210</font></i></a>, <a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></i>
;<br/>
<i><font color="Green" title="E80">A212</font></i>: <font color="Maroon" title="c52">Y3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">fy2</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c43">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c46">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14" title="PARTFUN3:func.14">/</a> <font color="Maroon" title="c43">m</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="#E55:22"><i><font color="Green" title="E35">A133</font></i></a>, <a class="txt" href="#E8:22_18"><i><font color="Green" title="E44">A176</font></i></a>, <a class="ref" href="valued_1.html#T1" title="VALUED_1:th.1">VALUED_1:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c24">xo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c26">dx</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c25">yo</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">OK</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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="c43">m</font> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <font color="Maroon" title="c46">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14" title="PARTFUN3:func.14">/</a> <font color="Maroon" title="c43">m</font></span>)</span> <a href="partfun3.html#K8" title="PARTFUN3:func.8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c27">dy</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c8">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="#E56:22"><i><font color="Green" title="E36">A134</font></i></a>, <a class="txt" href="#E3:22_18"><i><font color="Green" title="E40">A172</font></i></a>, <a class="txt" href="#E4:22_18"><i><font color="Green" title="E41">A173</font></i></a>, <a class="txt" href="#E6:22_18"><i><font color="Green" title="E43">A175</font></i></a>, <a class="txt" href="#E15:22_18"><i><font color="Green" title="E49">A181</font></i></a>, <a class="txt" href="#E16:22_18"><i><font color="Green" title="E50">A182</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c57">z</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c2">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#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E23:22_18"><i><font color="Green" title="E55">A187</font></i></a>, <a class="txt" href="#E24:22_18"><i><font color="Green" title="E56">A188</font></i></a>, <a class="txt" href="#E37:22_18"><i><font color="Green" title="E67">A199</font></i></a>, <a class="txt" href="#E50:22_18"><i><font color="Green" title="E78">A210</font></i></a>, <a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></i>
;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c10">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="c10">f</font>
<b>by </b><i><a class="ref" href="brouwer.html#D5" target="_self" title="BROUWER:def.5">Def5</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</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#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c2">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#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K21" title="EUCLID:func.21">`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="c3">o</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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="c3">o</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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="c2">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#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`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#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
<b>by </b><i><a class="txt" href="#E3:22_18"><i><font color="Green" title="E40">A172</font></i></a>, <a class="txt" href="#E4:22_18"><i><font color="Green" title="E41">A173</font></i></a>, <a class="txt" href="#E5:22_18"><i><font color="Green" title="E42">A174</font></i></a>, <a class="txt" href="#E6:22_18"><i><font color="Green" title="E43">A175</font></i></a>, <a class="ref" href="brouwer.html#T8" target="_self" title="BROUWER:th.8">Th8</a></i>
<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="c51">X3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c52">Y3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="#E51:22_18"><i><font color="Green" title="E79">A211</font></i></a>, <a class="txt" href="#E52:22_18"><i><font color="Green" title="E80">A212</font></i></a>, <a class="ref" href="topreala.html#D2" title="TOPREALA:def.2">TOPREALA:def 2</a></i>
<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="c51">X3</font>,<font color="Maroon" title="c52">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">y</font>,<font color="Maroon" title="c57">z</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="#E55:22"><i><font color="Green" title="E35">A133</font></i></a>, <a class="txt" href="#E8:22_18"><i><font color="Green" title="E44">A176</font></i></a>, <a class="txt" href="#E33:22_18"><i><font color="Green" title="E63">A195</font></i></a>, <a class="txt" href="#E34:22_18"><i><font color="Green" title="E64">A196</font></i></a>, <a class="ref" href="funct_3.html#T69" title="FUNCT_3:th.69">FUNCT_3:69</a></i>
<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="c51">X3</font>,<font color="Maroon" title="c52">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c55">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">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#K6" title="BORSUK_1:func.6">]</a></span>
<b>by </b><i><a class="txt" href="#E55:22"><i><font color="Green" title="E35">A133</font></i></a>, <a class="txt" href="#E3:22_18"><i><font color="Green" title="E40">A172</font></i></a>, <a class="txt" href="#E4:22_18"><i><font color="Green" title="E41">A173</font></i></a>, <a class="txt" href="#E8:22_18"><i><font color="Green" title="E44">A176</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E64:22"/>
( <font color="Maroon" title="c51">X3</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c52">Y3</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )
 
<b>by </b><i><a class="ref" href="topreal6.html#T83" title="TOPREAL6:th.83">TOPREAL6:83</a></i>;<br/>
<b>then reconsider </b><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="c51">X3</font>,<font color="Maroon" title="c52">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">OK</font></span>)</span>,<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> <b>by </b><i><a class="ref" href="yellow12.html#T41" title="YELLOW12:th.41">YELLOW12:41</a></i>;<br/>
<a NAME="E66:22"/>
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="c3">o</font>,<font color="Maroon" title="c2">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="c3">o</font>,<font color="Maroon" title="c2">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="c10">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="c3">o</font>,<font color="Maroon" title="c2">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="c10">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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"><b>proof </b></a><div class="add">

<b>let </b><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="c3">o</font>,<font color="Maroon" title="c2">r</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c3">o</font>,<font color="Maroon" title="c2">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="c10">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="c3">o</font>,<font color="Maroon" title="c2">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="c10">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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/><b>let </b><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="c3">o</font>,<font color="Maroon" title="c2">r</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c10">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="c3">o</font>,<font color="Maroon" title="c2">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="c10">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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/>



<b>assume </b><b>that </b><br/><a NAME="E1:22_19"/><i><font color="Green" title="E39">A213</font></i>: 
<span class="p1">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c10">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>
 <b>and </b><br/><a NAME="E2:22_19"/><i><font color="Green" title="E40">A214</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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="c3">o</font>,<font color="Maroon" title="c2">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="c10">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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/>

<b>reconsider </b><font color="Maroon" title="c58">p1</font> = <font color="Maroon" title="c56">p</font>, <font color="Maroon" title="c59">fp</font> = <font color="Maroon" title="c10">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> <b>by </b><i><a class="ref" href="pre_topc.html#T55" title="PRE_TOPC:th.55">PRE_TOPC:55</a></i>;<br/>
<a NAME="E4:22_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="c10">f</font>
 
<b>by </b><i><a class="txt" href="#E15:22"><i><font color="Green" title="E22">A10</font></i></a>, <a class="ref" href="abian.html#D5" title="ABIAN:def.5">ABIAN:def 5</a></i>;<br/>
<a NAME="E5:22_19"/><b>then </b>
<font color="Maroon" title="c56">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c56">p</font>
 
<b>by </b><i><a class="ref" href="abian.html#D4" title="ABIAN:def.4">ABIAN:def 4</a></i>;<br/>
<a NAME="E6:22_19"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c58">p1</font>,<font color="Maroon" title="c59">fp</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="E7:22_19"/><b>then </b><i><font color="Green" title="E41">A215</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">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#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">OK</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c60">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><b> such that </b><br/><a NAME="E9:22_19"/><i><font color="Green" title="E42">A216</font></i>: 
<font color="Maroon" title="c60">V1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E10:22_19"/><i><font color="Green" title="E43">A217</font></i>: 
<font color="Maroon" title="c60">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="c3">o</font>,<font color="Maroon" title="c2">r</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c57">V</font>
 <b>by </b><i><a class="txt" href="#E2:22_19"><i><font color="Green" title="E40">A214</font></i></a>, <a class="ref" href="tops_2.html#T32" title="TOPS_2:th.32">TOPS_2:32</a></i>;<br/>
<a NAME="E11:22_19"/><i><font color="Green" title="E44">A218</font></i>: 
<span class="p1">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c10">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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c56">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">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#K6" title="BORSUK_1:func.6">]</a></span>
 
<b>by </b><i><a class="txt" href="#E63:22"><i><font color="Green" title="E38">A170</font></i></a></i>;<br/>
<a NAME="E12:22_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>
 
<b>by </b><i><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></i>;<br/>
<a NAME="E13:22_19"/><b>then </b><i><font color="Green" title="E45">A219</font></i>: 
<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#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c60">V1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="#E9:22_19"><i><font color="Green" title="E42">A216</font></i></a>, <a class="ref" href="topgrp_1.html#T25" title="TOPGRP_1:th.25">TOPGRP_1:25</a></i>;<br/>
<a NAME="E14:22_19"/><i><font color="Green" title="E46">A220</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">OK</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E15:22_19"/><i><font color="Green" title="E47">A221</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E16:22_19"/><b>then </b><i><font color="Green" title="E48">A222</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">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#K4" title="RELSET_1:func.4">dom</a> <a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a> 
 
;<br/>
<a NAME="E17:22_19"/><b>then </b><i><font color="Green" title="E49">A223</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c55">F</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E18:22_19"/><i><font color="Green" title="E50">A224</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">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> &amp;  <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> )
 
<b>by </b><i><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></i>;<br/>
<i><font color="Green" title="E51">A225</font></i>: <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>
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
<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#K4" title="RELSET_1:func.4">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>
<b>by </b><i><a class="txt" href="#E18:22_19"><i><font color="Green" title="E50">A224</font></i></a>, <a class="ref" href="tops_2.html#T65" title="TOPS_2:th.65">TOPS_2:65</a></i>
;<br/>
<a NAME="E20:22_19"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">dom</a> <a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></i>;<br/>
<a NAME="E21:22_19"/><b>then </b><i><font color="Green" title="E52">A226</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c55">F</font>
 
<b>by </b><i><a class="txt" href="#E16:22_19"><i><font color="Green" title="E48">A222</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E22:22_19"/>
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#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">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_2"><b>proof </b></a><div class="add">

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">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/>

<b>assume </b><a NAME="E1:22_19_2"/><i><font color="Green" title="E53">A227</font></i>: 
<font color="Maroon" title="c61">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c55">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p3">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">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_2"/><i><font color="Green" title="E54">A228</font></i>: 
<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#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c55">F</font>
 
<b>by </b><i><a class="txt" href="#E1:22_19_2"><i><font color="Green" title="E53">A227</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>thus </b><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#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">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>
<b>by </b><i><a class="txt" href="#E1:22_19_2"><i><font color="Green" title="E53">A227</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<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>
<b>by </b><i><a class="txt" href="#E15:22_19"><i><font color="Green" title="E47">A221</font></i></a>, <a class="txt" href="#E2:22_19_2"><i><font color="Green" title="E54">A228</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E23:22_19"/><b>then </b><i><font color="Green" title="E53">A229</font></i>: 
<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#K4" title="RELSET_1:func.4">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>
 
<b>by </b><i><a class="txt" href="#E21:22_19"><i><font color="Green" title="E52">A226</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<a NAME="E24:22_19"/>
<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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c58">p1</font>,<font color="Maroon" title="c59">fp</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c60">V1</font>
 
<b>by </b><i><a class="txt" href="#E1:22_19"><i><font color="Green" title="E39">A213</font></i></a>, <a class="txt" href="#E10:22_19"><i><font color="Green" title="E43">A217</font></i></a>, <a class="txt" href="#E11:22_19"><i><font color="Green" title="E44">A218</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E25:22_19"/><b>then </b>
<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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c58">p1</font>,<font color="Maroon" title="c59">fp</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c60">V1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>
<a NAME="E26:22_19"/><b>then </b>
<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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c58">p1</font>,<font color="Maroon" title="c59">fp</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c60">V1</font>
 
<b>by </b><i><a class="txt" href="#E55:22"><i><font color="Green" title="E35">A133</font></i></a>, <a class="txt" href="#E7:22_19"><i><font color="Green" title="E41">A215</font></i></a>, <a class="txt" href="#E14:22_19"><i><font color="Green" title="E46">A220</font></i></a>, <a class="txt" href="#E17:22_19"><i><font color="Green" title="E49">A223</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/>
<b>then consider </b><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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">OK</font></span>)</span><b> such that </b><br/><a NAME="E28:22_19"/><i><font color="Green" title="E54">A230</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c58">p1</font>,<font color="Maroon" title="c59">fp</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c61">W</font>
 <b>and </b><br/><a NAME="E29:22_19"/><i><font color="Green" title="E55">A231</font></i>: 
<font color="Maroon" title="c61">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E30:22_19"/><i><font color="Green" title="E56">A232</font></i>: 
<font color="Maroon" title="c55">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c60">V1</font>
 <b>by </b><i><a class="txt" href="#E55:22"><i><font color="Green" title="E35">A133</font></i></a>, <a class="txt" href="#E7:22_19"><i><font color="Green" title="E41">A215</font></i></a>, <a class="txt" href="#E13:22_19"><i><font color="Green" title="E45">A219</font></i></a>, <a class="txt" href="#E19:22_19"><i><font color="Green" title="E51">A225</font></i></a>, <a class="txt" href="#E23:22_19"><i><font color="Green" title="E53">A229</font></i></a>, <a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></i>;<br/>
<b>consider </b><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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span><b> such that </b><br/><a NAME="E32:22_19"/><i><font color="Green" title="E57">A233</font></i>: 
<font color="Maroon" title="c62">WW</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E33:22_19"/><i><font color="Green" title="E58">A234</font></i>: 
<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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">OK</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c61">W</font>
 <b>by </b><i><a class="txt" href="#E29:22_19"><i><font color="Green" title="E55">A231</font></i></a>, <a class="ref" href="tops_2.html#T32" title="TOPS_2:th.32">TOPS_2:32</a></i>;<br/>
<b>consider </b><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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span><b> such that </b><br/><a NAME="E35:22_19"/><i><font color="Green" title="E59">A235</font></i>: 
<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>
 <b>and </b><br/><a NAME="E36:22_19"/><i><font color="Green" title="E60">A236</font></i>: 
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#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b2">X1</font>,<font color="Olive" title="b3">Y1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</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> )
 <b>by </b><i><a class="txt" href="#E32:22_19"><i><font color="Green" title="E57">A233</font></i></a>, <a class="ref" href="borsuk_1.html#T45" title="BORSUK_1:th.45">BORSUK_1:45</a></i>;<br/>
<a NAME="E37:22_19"/>
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c58">p1</font>,<font color="Maroon" title="c59">fp</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c62">WW</font>
 
<b>by </b><i><a class="txt" href="#E28:22_19"><i><font color="Green" title="E54">A230</font></i></a>, <a class="txt" href="#E33:22_19"><i><font color="Green" title="E58">A234</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c64">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E39:22_19"/><i><font color="Green" title="E61">A237</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c58">p1</font>,<font color="Maroon" title="c59">fp</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c64">Z</font>
 <b>and </b><br/><a NAME="E40:22_19"/><i><font color="Green" title="E62">A238</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="#E35:22_19"><i><font color="Green" title="E59">A235</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c65">X1</font>, <font color="Maroon" title="c66">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><b> such that </b><br/><a NAME="E42:22_19"/><i><font color="Green" title="E63">A239</font></i>: 
<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#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c65">X1</font>,<font color="Maroon" title="c66">Y1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>and </b><br/><a NAME="E43:22_19"/><i><font color="Green" title="E64">A240</font></i>: 
<font color="Maroon" title="c65">X1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E44:22_19"/><i><font color="Green" title="E65">A241</font></i>: 
<font color="Maroon" title="c66">Y1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="#E36:22_19"><i><font color="Green" title="E60">A236</font></i></a>, <a class="txt" href="#E40:22_19"><i><font color="Green" title="E62">A238</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c67">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">OK</font></span>)</span></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c68">XX</font> = <font color="Maroon" title="c65">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="c3">o</font>,<font color="Maroon" title="c2">r</font></span>)</span></span>)</span>, <font color="Maroon" title="c69">YY</font> = <font color="Maroon" title="c66">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="c3">o</font>,<font color="Maroon" title="c2">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="c3">o</font>,<font color="Maroon" title="c2">r</font></span>)</span> <b>by </b><i><a class="txt" href="#E43:22_19"><i><font color="Green" title="E64">A240</font></i></a>, <a class="txt" href="#E44:22_19"><i><font color="Green" title="E65">A241</font></i></a>, <a class="ref" href="tops_2.html#T32" title="TOPS_2:th.32">TOPS_2:32</a></i>;<br/>
<a NAME="E46:22_19"/>
<font color="Maroon" title="c59">fp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c66">Y1</font>
 
<b>by </b><i><a class="txt" href="#E39:22_19"><i><font color="Green" title="E61">A237</font></i></a>, <a class="txt" href="#E42:22_19"><i><font color="Green" title="E63">A239</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E47:22_19"/><b>then </b>
<font color="Maroon" title="c59">fp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c69">YY</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><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="c3">o</font>,<font color="Maroon" title="c2">r</font></span>)</span><b> such that </b><br/><a NAME="E49:22_19"/><i><font color="Green" title="E66">A242</font></i>: 
<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>
 <b>and </b><br/><a NAME="E50:22_19"/><i><font color="Green" title="E67">A243</font></i>: 
<font color="Maroon" title="c70">M</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E51:22_19"/><i><font color="Green" title="E68">A244</font></i>: 
<font color="Maroon" title="c10">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></i>;<br/>
<b>take </b><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:()"><i><font color="Red">::  thesis: </font></i></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="c10">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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="E52:22_19"/>
<font color="Maroon" title="c56">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c65">X1</font>
 
<b>by </b><i><a class="txt" href="#E39:22_19"><i><font color="Green" title="E61">A237</font></i></a>, <a class="txt" href="#E42:22_19"><i><font color="Green" title="E63">A239</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E53:22_19"/><b>then </b>
<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>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="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="c71">W1</font>
 <b>by </b><i><a class="txt" href="#E49:22_19"><i><font color="Green" title="E66">A242</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="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="c10">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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/>

<b>thus </b><a NAME="E55:22_19"/>
<font color="Maroon" title="c71">W1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="#E50:22_19"><i><font color="Green" title="E67">A243</font></i></a>, <a class="ref" href="tops_1.html#T38" title="TOPS_1:th.38">TOPS_1:38</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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/>

<b>let </b><font color="Maroon" title="c72">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="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="c10">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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/>

<b>assume </b><a NAME="E56:22_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="c10">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c71">W1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>then consider </b><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="c3">o</font>,<font color="Maroon" title="c2">r</font></span>)</span><b> such that </b><br/><a NAME="E58:22_19"/><i><font color="Green" title="E69">A245</font></i>: 
<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>
 <b>and </b><br/><a NAME="E59:22_19"/><i><font color="Green" title="E70">A246</font></i>: 
<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="c10">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c73">a</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<b>reconsider </b><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="c10">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> <b>by </b><i><a class="ref" href="pre_topc.html#T55" title="PRE_TOPC:th.55">PRE_TOPC:55</a></i>;<br/>
<a NAME="E61:22_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>
 
<b>by </b><i><a class="txt" href="#E58:22_19"><i><font color="Green" title="E69">A245</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E62:22_19"/><b>then </b><i><font color="Green" title="E71">A247</font></i>: 
<font color="Maroon" title="c73">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c65">X1</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E63:22_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="c10">f</font>
 
<b>by </b><i><a class="txt" href="#E15:22"><i><font color="Green" title="E22">A10</font></i></a>, <a class="ref" href="abian.html#D5" title="ABIAN:def.5">ABIAN:def 5</a></i>;<br/>
<a NAME="E64:22_19"/><b>then </b>
<font color="Maroon" title="c73">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c73">a</font>
 
<b>by </b><i><a class="ref" href="abian.html#D4" title="ABIAN:def.4">ABIAN:def 4</a></i>;<br/>
<a NAME="E65:22_19"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c74">a1</font>,<font color="Maroon" title="c75">fa</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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="E66:22_19"/><b>then </b><i><font color="Green" title="E72">A248</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c73">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">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#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">OK</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E67:22_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>
 
<b>by </b><i><a class="txt" href="#E58:22_19"><i><font color="Green" title="E69">A245</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E68:22_19"/><b>then </b>
<font color="Maroon" title="c75">fa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c70">M</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>
<a NAME="E69:22_19"/><b>then </b>
<font color="Maroon" title="c10">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="c66">Y1</font>
 
<b>by </b><i><a class="txt" href="#E51:22_19"><i><font color="Green" title="E68">A244</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E70:22_19"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c73">a</font>,<font color="Maroon" title="c75">fa</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c64">Z</font>
 
<b>by </b><i><a class="txt" href="#E42:22_19"><i><font color="Green" title="E63">A239</font></i></a>, <a class="txt" href="#E62:22_19"><i><font color="Green" title="E71">A247</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E71:22_19"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c73">a</font>,<font color="Maroon" title="c75">fa</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">OK</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E55:22"><i><font color="Green" title="E35">A133</font></i></a>, <a class="txt" href="#E66:22_19"><i><font color="Green" title="E72">A248</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E72:22_19"/><b>then </b><i><font color="Green" title="E73">A249</font></i>: 
<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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c74">a1</font>,<font color="Maroon" title="c75">fa</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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#K10" title="RELSET_1:func.10">.:</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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">OK</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>
<a NAME="E73:22_19"/><i><font color="Green" title="E74">A250</font></i>: 
<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> 
 
<b>by </b><i><a class="txt" href="#E18:22_19"><i><font color="Green" title="E50">A224</font></i></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></i>;<br/>
<a NAME="E74:22_19"/><i><font color="Green" title="E75">A251</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">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>
 
<b>by </b><i><a class="txt" href="#E18:22_19"><i><font color="Green" title="E50">A224</font></i></a>, <a class="ref" href="tops_2.html#T62" title="TOPS_2:th.62">TOPS_2:62</a></i>;<br/>
<a NAME="E75:22_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>
 
<b>by </b><i><a class="txt" href="#E35:22_19"><i><font color="Green" title="E59">A235</font></i></a>, <a class="txt" href="#E40:22_19"><i><font color="Green" title="E62">A238</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E76:22_19"/><b>then </b>
<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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">OK</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></i>;<br/>
<a NAME="E77:22_19"/><b>then </b>
<font color="Maroon" title="c55">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">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#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c61">W</font>
 
<b>by </b><i><a class="txt" href="#E33:22_19"><i><font color="Green" title="E58">A234</font></i></a>, <a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a></i>;<br/>
<a NAME="E78:22_19"/><b>then </b>
<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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c74">a1</font>,<font color="Maroon" title="c75">fa</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c61">W</font>
 
<b>by </b><i><a class="txt" href="#E72:22_19"><i><font color="Green" title="E73">A249</font></i></a></i>;<br/>
<a NAME="E79:22_19"/><b>then </b>
<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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c74">a1</font>,<font color="Maroon" title="c75">fa</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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#K10" title="RELSET_1:func.10">.:</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#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c60">V1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E30:22_19"><i><font color="Green" title="E56">A232</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>
<a NAME="E80:22_19"/><b>then </b>
<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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c74">a1</font>,<font color="Maroon" title="c75">fa</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</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#K10" title="RELSET_1:func.10">.:</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#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c60">V1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E55:22"><i><font color="Green" title="E35">A133</font></i></a>, <a class="txt" href="#E66:22_19"><i><font color="Green" title="E72">A248</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>;<br/>
<a NAME="E81:22_19"/><b>then </b>
<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#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c74">a1</font>,<font color="Maroon" title="c75">fa</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c60">V1</font>
 
<b>by </b><i><a class="txt" href="#E18:22_19"><i><font color="Green" title="E50">A224</font></i></a>, <a class="txt" href="#E73:22_19"><i><font color="Green" title="E74">A250</font></i></a>, <a class="txt" href="#E74:22_19"><i><font color="Green" title="E75">A251</font></i></a>, <a class="ref" href="partfun3.html#T1" title="PARTFUN3:th.1">PARTFUN3:1</a></i>;<br/>
<a NAME="E82:22_19"/><b>then </b>
<span class="p1">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c10">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="c60">V1</font>
 
<b>by </b><i><a class="txt" href="#E63:22"><i><font color="Green" title="E38">A170</font></i></a></i>;<br/>
<b>hence </b><a NAME="E83:22_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>
 <b>by </b><i><a class="txt" href="#E10:22_19"><i><font color="Green" title="E43">A217</font></i></a>, <a class="txt" href="#E59:22_19"><i><font color="Green" title="E70">A246</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E67:22"/>
 <a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c10">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
