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

<b>let </b><font color="Maroon" title="c2">F</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ;<br/>

<b>assume </b><a NAME="E1:17"/><i><font color="Green" title="E14">A1</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 <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <br/> for <font color="Olive" title="b2">i</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds  <font color="Maroon" title="c2">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">x</font>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span> = <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> $2</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> $1;<br/>
<b>consider </b><font color="Maroon" title="c3">G</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,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> 1</span>)</span><b> such that </b><br/><a NAME="E3:17"/><i><font color="Green" title="E15">A2</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="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span><br/> for <font color="Olive" title="b2">i</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds  <font color="Maroon" title="c3">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">i</font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4" title="BINOP_1:sch.4">BINOP_1:sch 4</a>();<br/></i>
<b>reconsider </b><font color="Maroon" title="c4">G</font> = <font color="Maroon" title="c3">G</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <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="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>,<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span> <b>by </b><i><a class="txt" href="topalg_2.html#E4"><i><font color="Green" title="E4">Lm2</font></i></a></i>;<br/>
<a NAME="E5:17"/><i><font color="Green" title="E16">A3</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="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span><br/> for <font color="Olive" title="b2">i</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds  <font color="Maroon" title="c4">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">i</font>)
 
<b>by </b><i><a class="txt" href="topalg_2.html#E3:17"><i><font color="Green" title="E15">A2</font></i></a></i>;<br/>
<a NAME="E6:17"/><i><font color="Green" title="E17">A4</font></i>: 
<font color="Maroon" title="c4">G</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="topalg_2.html#E5:17"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="topalg_1.html#T18" title="TOPALG_1:th.18">TOPALG_1:18</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <b> such that </b><br/><a NAME="E8:17"/><i><font color="Green" title="E18">A5</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span> holds  <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan2b.html#K1" title="JORDAN2B:func.1">Proj</a> <font color="Olive" title="b1">p</font>,1
 <b>by </b><i><a class="ref" href="jordan2b.html#T1" title="JORDAN2B:th.1">JORDAN2B:1</a></i>;<br/>
<a NAME="E9:17"/><i><font color="Green" title="E19">A6</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 
<b>by </b><i><a class="txt" href="topalg_2.html#E8:17"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="jordan2b.html#T34" title="JORDAN2B:th.34">JORDAN2B:34</a></i>;<br/>
<a NAME="E10:17"/><b>then </b><i><font color="Green" title="E20">A7</font></i>: 
<font color="Maroon" title="c5">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E11:17"/>
<font color="Maroon" title="c5">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="topalg_2.html#E9:17"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E12:17"/><b>then </b>
<span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span>,<span class="p2">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="borsuk_2.html#T12" title="BORSUK_2:th.12">BORSUK_2:12</a></i>;<br/>
<a NAME="E13:17"/><b>then </b>
<font color="Maroon" title="c4">G</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span>,<span class="p2">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="topalg_2.html#E6:17"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="tops_2.html#T58" title="TOPS_2:th.58">TOPS_2:58</a></i>;<br/>
<a NAME="E14:17"/><b>then </b><i><font color="Green" title="E21">A8</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span>,<span class="p3">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span></span>)</span> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="topalg_2.html#E10:17"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="tops_2.html#T58" title="TOPS_2:th.58">TOPS_2:58</a></i>;<br/>
<a NAME="E15:17"/>
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#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> holds  <font color="Maroon" title="c2">F</font> <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"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span>,<span class="p4">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>;<br/>

<b>consider </b><font color="Maroon" title="c7">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> , <font color="Maroon" title="c8">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <b> such that </b><br/><a NAME="E2:17_1"/><i><font color="Green" title="E22">A9</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</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="E3:17_1"/><i><font color="Green" title="E23">A10</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">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"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</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="E4:17_1"/><i><font color="Green" title="E24">A11</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </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 <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> 
 
<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>reconsider </b><font color="Maroon" title="c9">ff</font> = <font color="Maroon" title="c5">f</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<b>set </b><font color="Maroon" title="c10">g</font> = <font color="Maroon" title="c9">ff</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> ;<br/>
<a NAME="E6:17_1"/><i><font color="Green" title="E25">A12</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> 
 
<b>by </b><i><a class="txt" href="topalg_2.html#E9:17"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E7:17_1"/><i><font color="Green" title="E26">A13</font></i>: 
<font color="Maroon" title="c9">ff</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="topalg_2.html#E9:17"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E8:17_1"/><i><font color="Green" title="E27">A14</font></i>: 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span>
 
<b>by </b><i><a class="ref" href="jordan2b.html#D2" title="JORDAN2B:def.2">JORDAN2B:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">z</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E10:17_1"/><i><font color="Green" title="E28">A15</font></i>: 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">z</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="jordan2b.html#T24" title="JORDAN2B:th.24">JORDAN2B:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">w</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1 <b>by </b><i><a class="txt" href="topalg_2.html#E8:17_1"><i><font color="Green" title="E27">A14</font></i></a>, <a class="ref" href="euclid.html#T25" title="EUCLID:th.25">EUCLID:25</a></i>;<br/>
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> = 
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span>
<b>by </b><i><a class="ref" href="jordan2b.html#D2" title="JORDAN2B:def.2">JORDAN2B:def 2</a></i>
<br/>.= 
 <a href="jordan2b.html#K1" title="JORDAN2B:func.1">Proj</a> <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span>,1
<b>by </b><i><a class="txt" href="topalg_2.html#E8:17"><i><font color="Green" title="E18">A5</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c7">a</font>
<b>by </b><i><a class="ref" href="topalg_2.html#T4" target="_self" title="TOPALG_2:th.4">Th4</a></i>
;<br/>
<a NAME="E13:17_1"/><b>then </b>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">ff</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font>
 
<b>by </b><i><a class="txt" href="topalg_2.html#E3:17_1"><i><font color="Green" title="E23">A10</font></i></a>, <a class="txt" href="topalg_2.html#E7:17_1"><i><font color="Green" title="E26">A13</font></i></a>, <a class="txt" href="topalg_2.html#E8:17_1"><i><font color="Green" title="E27">A14</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>;<br/>
<a NAME="E14:17_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="topalg_2.html#E6:17_1"><i><font color="Green" title="E25">A12</font></i></a>, <a class="txt" href="topalg_2.html#E7:17_1"><i><font color="Green" title="E26">A13</font></i></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></i>;<br/>
<b>then </b><i><font color="Green" title="E29">A16</font></i>: <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">z</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> = 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c12">w</font>
<b>by </b><i><a class="txt" href="topalg_2.html#E10:17_1"><i><font color="Green" title="E28">A15</font></i></a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">a</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T69" title="RVSUM_1:th.69">RVSUM_1:69</a></i>
;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span>,<span class="p4">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font> = 
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span>,<span class="p4">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span>,<span class="p4">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="topalg_2.html#E2:17_1"><i><font color="Green" title="E22">A9</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="topalg_2.html#E4:17_1"><i><font color="Green" title="E24">A11</font></i></a>, <a class="txt" href="topalg_2.html#E6"><i><font color="Green" title="E6">Lm4</font></i></a>, <a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<font color="Maroon" title="c8">b</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="topalg_2.html#E5:17"><i><font color="Green" title="E16">A3</font></i></a></i>
<br/>.= 
 <a href="jordan2b.html#K1" title="JORDAN2B:func.1">Proj</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span></span>)</span>,1
<b>by </b><i><a class="txt" href="topalg_2.html#E8:17"><i><font color="Green" title="E18">A5</font></i></a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">z</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1
<b>by </b><i><a class="txt" href="topalg_2.html#E10:17_1"><i><font color="Green" title="E28">A15</font></i></a>, <a class="ref" href="jordan2b.html#D1" title="JORDAN2B:def.1">JORDAN2B:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c11">z</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T25" title="FINSEQ_4:th.25">FINSEQ_4:25</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">a</font>
<b>by </b><i><a class="txt" href="topalg_2.html#E15:17_1"><i><font color="Green" title="E29">A16</font></i></a>, <a class="ref" href="group_7.html#T1" title="GROUP_7:th.1">GROUP_7:1</a></i>
<br/>.= 
<font color="Maroon" title="c2">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>
<b>by </b><i><a class="txt" href="topalg_2.html#E1:17"><i><font color="Green" title="E14">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font>
<b>by </b><i><a class="txt" href="topalg_2.html#E2:17_1"><i><font color="Green" title="E22">A9</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E16:17"/>
<font color="Maroon" title="c2">F</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="topalg_2.html#E14:17"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>


</div>
