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<div class="add">

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:60_2_1"/><i><font color="Green" title="E43">A7</font></i>: 
( <font color="Maroon" title="c1">T</font> <a href="urysohn1.html#NR1" title="URYSOHN1:NR.1">is_T1</a>  &amp; <font color="Maroon" title="c1">T</font> <a href="compts_1.html#NR4" title="COMPTS_1:NR.4">is_T4</a>  )
 ;<br/>

<b>hence </b><a NAME="E2:60_2_1"/>
<font color="Maroon" title="c1">T</font> <a href="urysohn1.html#NR1" title="URYSOHN1:NR.1">is_T1</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="topgen_5.html#D4" target="_self" title="TOPGEN_5:def.4">TOPGEN_5:def 4</a><br/>

<b>let </b><font color="Maroon" title="c2">A</font> be  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/><b>let </b><font color="Maroon" title="c3">a</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>;<br/>



<a NAME="E3:60_2_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">j</font> = 1, <font color="Maroon" title="c5">k</font> =  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> <b>by </b><i><a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">z</font> = 0, <font color="Maroon" title="c7">o</font> = 1 as   <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>by </b><i><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:60_2_1_1"/>
( <font color="Maroon" title="c2">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Maroon" title="c2">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_2_1_1_1"><b>suppose </b></a><a NAME="E1:60_2_1_1_1"/><i><font color="Green" title="E44">A8</font></i>: 
<font color="Maroon" title="c2">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c1">T</font> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c6">z</font>;<br/><a NAME="E2:60_2_1_1_1"/>
( <font color="Maroon" title="c1">T</font> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c6">z</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="borsuk_1.html#K1" title="BORSUK_1:func.1">--&gt;</a> <font color="Maroon" title="c6">z</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c6">z</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:60_2_1_1_1"><i><font color="Green" title="E44">A8</font></i></a>, <a class="ref" href="relat_1.html#T149" title="RELAT_1:th.149">RELAT_1:149</a></i>;<br/><a NAME="E3:60_2_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c6">z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c6">z</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E4:60_2_1_1_1"/>
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  implies  ex <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 <font color="Maroon" title="c1">T</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <br/>( <font color="Olive" title="b1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Olive" title="b1">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_2_1_1_2"><b>suppose </b></a><a NAME="E1:60_2_1_1_2"/>
not <font color="Maroon" title="c2">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c8">aa</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>, <font color="Maroon" title="c9">A'</font> = <font color="Maroon" title="c2">A</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="topgen_5.html#E1:60_2_1"><i><font color="Green" title="E43">A7</font></i></a>, <a class="ref" href="urysohn1.html#T27" title="URYSOHN1:th.27">URYSOHN1:27</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">B</font> = <font color="Maroon" title="c9">A'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">aa</font> as  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="tops_1.html#T36" title="TOPS_1:th.36">TOPS_1:36</a></i>;<br/><b>set </b><font color="Maroon" title="c11">h</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">A'</font></span>)</span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">aa</font></span>)</span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c5">k</font></span>)</span>;<br/><a NAME="E4:60_2_1_1_2"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">A'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">aa</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">aa</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="E5:60_2_1_1_2"/><b>then </b><i><font color="Green" title="E44">A9</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">A'</font></span>)</span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c4">j</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c9">A'</font> <a href="borsuk_1.html#K1" title="BORSUK_1:func.1">--&gt;</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">aa</font></span>)</span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c5">k</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c8">aa</font> <a href="borsuk_1.html#K1" title="BORSUK_1:func.1">--&gt;</a> <font color="Maroon" title="c5">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">A'</font> <a href="borsuk_1.html#K1" title="BORSUK_1:func.1">--&gt;</a> 1</span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="ref" href="borsuk_1.html#T6" title="BORSUK_1:th.6">BORSUK_1:6</a></i>;<br/><b>assume </b><a NAME="E6:60_2_1_1_2"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ;<br/><a NAME="E7:60_2_1_1_2"/><b>then </b>
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">A'</font></span>)</span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">aa</font></span>)</span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">k</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">k</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; not <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T17" title="ZFMISC_1:th.17">ZFMISC_1:17</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a>, <a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><a NAME="E8:60_2_1_1_2"/><b>then </b><i><font color="Green" title="E45">A10</font></i>: 
( <font color="Maroon" title="c9">A'</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> <font color="Maroon" title="c8">aa</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">A'</font></span>)</span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">aa</font></span>)</span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c5">k</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T64" title="XBOOLE_1:th.64">XBOOLE_1:64</a>, <a class="ref" href="zfmisc_1.html#T56" title="ZFMISC_1:th.56">ZFMISC_1:56</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c12">h</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">A'</font></span>)</span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">aa</font></span>)</span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c5">k</font></span>)</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"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c10">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> <b>by </b><i><a class="ref" href="topgen_5.html#T15" target="_self" title="TOPGEN_5:th.15">Th15</a></i>;<br/><a NAME="E10:60_2_1_1_2"/><i><font color="Green" title="E46">A11</font></i>: 
<font color="Maroon" title="c12">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="topgen_5.html#E5:60_2_1_1_2"><i><font color="Green" title="E44">A9</font></i></a>, <a class="txt" href="topgen_5.html#E8:60_2_1_1_2"><i><font color="Green" title="E45">A10</font></i></a>, <a class="ref" href="boolmark.html#T3" title="BOOLMARK:th.3">BOOLMARK:3</a></i>;<br/><a NAME="E11:60_2_1_1_2"/><i><font color="Green" title="E47">A12</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"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">aa</font></span>)</span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">aa</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">aa</font> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E5:60_2_1_1_2"><i><font color="Green" title="E44">A9</font></i></a>, <a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then </b><i><font color="Green" title="E48">A13</font></i>: <font color="Maroon" title="c12">h</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">a</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">aa</font></span>)</span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">a</font>
<b>by </b><i><a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
<b>by </b><i><a class="txt" href="topgen_5.html#E11:60_2_1_1_2"><i><font color="Green" title="E47">A12</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><a NAME="E13:60_2_1_1_2"/><i><font color="Green" title="E49">A14</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B</font> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E11:60_2_1_1_2"><i><font color="Green" title="E47">A12</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>consider </b><font color="Maroon" title="c13">g</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 <font color="Maroon" title="c1">T</font>,<span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span><b> such that </b><br/><a NAME="E15:60_2_1_1_2"/><i><font color="Green" title="E50">A15</font></i>: 
<font color="Maroon" title="c13">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">h</font>
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:60_2_1"><i><font color="Green" title="E43">A7</font></i></a>, <a class="ref" href="tietze.html#T25" title="TIETZE:th.25">TIETZE:25</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">p</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"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span><b> such that </b><br/><a NAME="E17:60_2_1_1_2"/><i><font color="Green" title="E51">A16</font></i>: 
( <font color="Maroon" title="c14">p</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; ( for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> holds <br/><font color="Maroon" title="c14">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 ) &amp; <font color="Maroon" title="c14">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <font color="Maroon" title="c14">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="jgraph_5.html#T42" title="JGRAPH_5:th.42">JGRAPH_5:42</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">p'</font> = <font color="Maroon" title="c14">p</font> <a href="tops_2.html#K2" title="TOPS_2:func.2">/"</a>  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"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="topgen_5.html#E17:60_2_1_1_2"><i><font color="Green" title="E51">A16</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">f</font> = <font color="Maroon" title="c15">p'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">g</font> 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 <font color="Maroon" title="c1">T</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="ref" href="tops_2.html#T58" title="TOPS_2:th.58">TOPS_2:58</a></i>;<br/><b>take </b>
<font color="Maroon" title="c16">f</font>
;<br/><a NAME="E20:60_2_1_1_2"/><i><font color="Green" title="E52">A17</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">p</font> <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="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> &amp; <font color="Maroon" title="c14">p</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="topgen_5.html#E17:60_2_1_1_2"><i><font color="Green" title="E51">A16</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="E21:60_2_1_1_2"/><b>then </b><i><font color="Green" title="E53">A18</font></i>: 
<font color="Maroon" title="c15">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">p</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
 <b>by </b><i><a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></i>;<br/><a NAME="E22:60_2_1_1_2"/><b>then </b><i><font color="Green" title="E54">A19</font></i>: 
<font color="Maroon" title="c15">p'</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z</font>
 <b>by </b><i><a class="txt" href="topgen_5.html#E17:60_2_1_1_2"><i><font color="Green" title="E51">A16</font></i></a>, <a class="txt" href="topgen_5.html#E20:60_2_1_1_2"><i><font color="Green" title="E52">A17</font></i></a>, <a class="ref" href="funct_2.html#T32" title="FUNCT_2:th.32">FUNCT_2:32</a></i>;<br/><b>thus </b><font color="Maroon" title="c16">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">a</font> = 
<font color="Maroon" title="c15">p'</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">a</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/>.= 
0
<b>by </b><i><a class="txt" href="topgen_5.html#E15:60_2_1_1_2"><i><font color="Green" title="E50">A15</font></i></a>, <a class="txt" href="topgen_5.html#E12:60_2_1_1_2"><i><font color="Green" title="E48">A13</font></i></a>, <a class="txt" href="topgen_5.html#E13:60_2_1_1_2"><i><font color="Green" title="E49">A14</font></i></a>, <a class="txt" href="topgen_5.html#E22:60_2_1_1_2"><i><font color="Green" title="E54">A19</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/><font color="Maroon" title="c16">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">A</font> = 
<font color="Maroon" title="c15">p'</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">A</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T159" title="RELAT_1:th.159">RELAT_1:159</a></i>
<br/>.= 
<font color="Maroon" title="c15">p'</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">A</font></span>)</span>
<b>by </b><i><a class="txt" href="topgen_5.html#E15:60_2_1_1_2"><i><font color="Green" title="E50">A15</font></i></a>, <a class="txt" href="topgen_5.html#E13:60_2_1_1_2"><i><font color="Green" title="E49">A14</font></i></a>, <a class="ref" href="relat_1.html#T162" title="RELAT_1:th.162">RELAT_1:162</a></i>
;<br/><a NAME="E25:60_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c16">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">p'</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c15">p'</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">o</font> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E10:60_2_1_1_2"><i><font color="Green" title="E46">A11</font></i></a>, <a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a>, <a class="txt" href="topgen_5.html#E21:60_2_1_1_2"><i><font color="Green" title="E53">A18</font></i></a>, <a class="txt" href="topgen_5.html#E17:60_2_1_1_2"><i><font color="Green" title="E51">A16</font></i></a>, <a class="txt" href="topgen_5.html#E20:60_2_1_1_2"><i><font color="Green" title="E52">A17</font></i></a>, <a class="ref" href="funct_2.html#T32" title="FUNCT_2:th.32">FUNCT_2:32</a></i>;<br/><b>hence </b><a NAME="E26:60_2_1_1_2"/>
<font color="Maroon" title="c16">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="setwiseo.html#T13" title="SETWISEO:th.13">SETWISEO:13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
