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

<span class="kw">let </span><font color="Maroon" title="c7">C</font> be   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">U</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">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>  st <font color="Olive" title="b1">U</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">U</font> is  <a href="borsuk_2.html#V1" title="BORSUK_2:attr.1">arcwise_connected</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c8">U</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">U</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">U</font> is  <a href="borsuk_2.html#V1" title="BORSUK_2:attr.1">arcwise_connected</a>  )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c9">T</font> = <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E1:101"/><span class="lab"><font color="Green" title="E75">A1</font></span>: 
<font color="Maroon" title="c8">U</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">U</font> is  <a href="borsuk_2.html#V1" title="BORSUK_2:attr.1">arcwise_connected</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c10">a</font>, <font color="Maroon" title="c11">b</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">U</font></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="borsuk_2.html#D3" title="BORSUK_2:def.3">BORSUK_2:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font> <a href="borsuk_2.html#R1" title="BORSUK_2:pred.1">are_connected</a> </span><br/>

<a NAME="E2:101"/><span class="lab"><font color="Green" title="E76">A2</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">U</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></span>;<br/>
<a NAME="E3:101"/><span class="lab"><font color="Green" title="E77">A3</font></span>: 
<font color="Maroon" title="c8">U</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101"><span class="lab"><font color="Green" title="E75">A1</font></span></a>, <a class="ref" href="connsp_1.html#T34" title="CONNSP_1:th.34">CONNSP_1:34</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:101_1"/>
( <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b</font> or <font color="Maroon" title="c10">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1"><span class="kw">suppose </span></a><a NAME="E1:101_1_1"/><span class="lab"><font color="Green" title="E78">A4</font></span>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font> <a href="borsuk_2.html#R1" title="BORSUK_2:pred.1">are_connected</a> </span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c12">TU</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">U</font> as  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101"><span class="lab"><font color="Green" title="E77">A3</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">a</font> = <font color="Maroon" title="c10">a</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c12">TU</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">f</font> = <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <font color="Maroon" title="c13">a</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">U</font></span>)</span> ;<br/><span class="kw">take </span>
<font color="Maroon" title="c14">f</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="borsuk_2.html#D1" title="BORSUK_2:def.1">BORSUK_2:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; <font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font> &amp; <font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b</font> )</span><br/><span class="kw">thus </span><a NAME="E5:101_1_1"/>
( <font color="Maroon" title="c14">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; <font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font> &amp; <font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_1"><span class="lab"><font color="Green" title="E78">A4</font></span></a>, <a class="ref" href="borsuk_1.html#D17" title="BORSUK_1:def.17">BORSUK_1:def 17</a>, <a class="ref" href="borsuk_1.html#D18" title="BORSUK_1:def.18">BORSUK_1:def 18</a>, <a class="ref" href="topalg_3.html#T5" title="TOPALG_3:th.5">TOPALG_3:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_2"><span class="kw">suppose </span></a><a NAME="E1:101_1_2"/><span class="lab"><font color="Green" title="E78">A5</font></span>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font> <a href="borsuk_2.html#R1" title="BORSUK_2:pred.1">are_connected</a> </span><br/><div class="add"><a NAME="E2:101_1_2"/><span class="lab"><font color="Green" title="E79">A6</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">U</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T7" title="TSEP_1:th.7">TSEP_1:7</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">a1</font> = <font color="Maroon" title="c10">a</font>, <font color="Maroon" title="c13">b1</font> = <font color="Maroon" title="c11">b</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101"><span class="lab"><font color="Green" title="E77">A3</font></span></a>, <a class="ref" href="pre_topc.html#T55" title="PRE_TOPC:th.55">PRE_TOPC:55</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">V</font> = <font color="Maroon" title="c8">U</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T39" title="PRE_TOPC:th.39">PRE_TOPC:39</a></span>;<br/><a NAME="E5:101_1_2"/>
<font color="Maroon" title="c14">V</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101"><span class="lab"><font color="Green" title="E75">A1</font></span></a>, <a class="ref" href="connsp_1.html#D6" title="CONNSP_1:def.6">CONNSP_1:def 6</a></span>;<br/><a NAME="E6:101_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A7</font></span>: 
<font color="Maroon" title="c14">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_3.html#T18" title="SPRECT_3:th.18">SPRECT_3:18</a></span>;<br/><a NAME="E7:101_1_2"/>
<font color="Maroon" title="c8">U</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101"><span class="lab"><font color="Green" title="E75">A1</font></span></a>, <a class="ref" href="connsp_1.html#D5" title="CONNSP_1:def.5">CONNSP_1:def 5</a></span>;<br/><a NAME="E8:101_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">V</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_1.html#T24" title="CONNSP_1:th.24">CONNSP_1:24</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">P</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E10:101_1_2"/><span class="lab"><font color="Green" title="E81">A8</font></span>: 
<font color="Maroon" title="c15">P</font> <a href="topreal4.html#R1" title="TOPREAL4:pred.1">is_S-P_arc_joining</a> <font color="Maroon" title="c12">a1</font>,<font color="Maroon" title="c13">b1</font>
 <span class="kw">and </span><br/><a NAME="E11:101_1_2"/><span class="lab"><font color="Green" title="E82">A9</font></span>: 
<font color="Maroon" title="c15">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101"><span class="lab"><font color="Green" title="E76">A2</font></span></a>, <a class="txt" href="#E3:101"><span class="lab"><font color="Green" title="E77">A3</font></span></a>, <a class="txt" href="#E1:101_1_2"><span class="lab"><font color="Green" title="E78">A5</font></span></a>, <a class="txt" href="#E6:101_1_2"><span class="lab"><font color="Green" title="E80">A7</font></span></a>, <a class="ref" href="topreal4.html#T30" title="TOPREAL4:th.30">TOPREAL4:30</a></span>;<br/><a NAME="E12:101_1_2"/><span class="lab"><font color="Green" title="E83">A10</font></span>: 
<font color="Maroon" title="c12">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:101_1_2"><span class="lab"><font color="Green" title="E81">A8</font></span></a>, <a class="ref" href="topreal4.html#T4" title="TOPREAL4:th.4">TOPREAL4:4</a></span>;<br/><a NAME="E13:101_1_2"/>
<font color="Maroon" title="c15">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c12">a1</font>,<font color="Maroon" title="c13">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:101_1_2"><span class="lab"><font color="Green" title="E81">A8</font></span></a>, <a class="ref" href="topreal4.html#T3" title="TOPREAL4:th.3">TOPREAL4:3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c15">P</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E15:101_1_2"/><span class="lab"><font color="Green" title="E84">A11</font></span>: 
<font color="Maroon" title="c16">g</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 <span class="kw">and </span><br/><a NAME="E16:101_1_2"/><span class="lab"><font color="Green" title="E85">A12</font></span>: 
<font color="Maroon" title="c16">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font>
 <span class="kw">and </span><br/><a NAME="E17:101_1_2"/><span class="lab"><font color="Green" title="E86">A13</font></span>: 
<font color="Maroon" title="c16">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#D2" title="TOPREAL1:def.2">TOPREAL1:def 2</a></span>;<br/><a NAME="E18:101_1_2"/><span class="lab"><font color="Green" title="E87">A14</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c15">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">f</font> = <font color="Maroon" title="c16">g</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">U</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101"><span class="lab"><font color="Green" title="E76">A2</font></span></a>, <a class="txt" href="#E11:101_1_2"><span class="lab"><font color="Green" title="E82">A9</font></span></a>, <a class="txt" href="#E12:101_1_2"><span class="lab"><font color="Green" title="E83">A10</font></span></a>, <a class="ref" href="funct_2.html#T9" title="FUNCT_2:th.9">FUNCT_2:9</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c17">f</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="borsuk_2.html#D1" title="BORSUK_2:def.1">BORSUK_2:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font> &amp; <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b</font> )</span><br/><a NAME="E20:101_1_2"/>
<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c15">P</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">U</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101"><span class="lab"><font color="Green" title="E76">A2</font></span></a>, <a class="txt" href="#E2:101_1_2"><span class="lab"><font color="Green" title="E79">A6</font></span></a>, <a class="txt" href="#E11:101_1_2"><span class="lab"><font color="Green" title="E82">A9</font></span></a>, <a class="txt" href="#E18:101_1_2"><span class="lab"><font color="Green" title="E87">A14</font></span></a>, <a class="ref" href="tsep_1.html#T4" title="TSEP_1:th.4">TSEP_1:4</a></span>;<br/><span class="kw">hence </span><a NAME="E21:101_1_2"/>
<font color="Maroon" title="c17">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:101_1_2"><span class="lab"><font color="Green" title="E84">A11</font></span></a>, <a class="ref" href="pre_topc.html#T56" title="PRE_TOPC:th.56">PRE_TOPC:56</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font> &amp; <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b</font> )</span><br/><span class="kw">thus </span><a NAME="E22:101_1_2"/>
( <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font> &amp; <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:101_1_2"><span class="lab"><font color="Green" title="E85">A12</font></span></a>, <a class="txt" href="#E17:101_1_2"><span class="lab"><font color="Green" title="E86">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
