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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</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> <font color="Maroon" title="c1">n</font></span>)</span><br/>  for <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</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> <font color="Maroon" title="c1">n</font></span>)</span><br/>  for <font color="Olive" title="b4">a</font>, <font color="Olive" title="b5">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>  st <font color="Olive" title="b1">A</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Olive" title="b2">p</font>,<font color="Olive" title="b3">q</font> &amp; <font color="Olive" title="b4">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">b</font> &amp; <font color="Olive" title="b5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 holds <br/> ex <font color="Olive" title="b6">E</font> being   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 <a href="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a> ex <font color="Olive" title="b7">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="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b6">E</font></span>)</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> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b2">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b6">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Olive" title="b4">a</font>,<font color="Olive" title="b5">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> &amp; <font color="Olive" title="b7">f</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>  &amp; <font color="Olive" title="b7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">q</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <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> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</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> <font color="Maroon" title="c1">n</font></span>)</span><br/>  for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">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>  st <font color="Maroon" title="c2">A</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> &amp; <font color="Olive" title="b3">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">b</font> &amp; <font color="Olive" title="b4">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 holds <br/> ex <font color="Olive" title="b5">E</font> being   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 <a href="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a> ex <font color="Olive" title="b6">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="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b5">E</font></span>)</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> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> &amp; <font color="Olive" title="b6">f</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>  &amp; <font color="Olive" title="b6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</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> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">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>  st <font color="Maroon" title="c2">A</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font> &amp; <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 holds <br/> ex <font color="Olive" title="b3">E</font> being   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 <a href="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a> ex <font color="Olive" title="b4">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="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b3">E</font></span>)</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> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b3">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> &amp; <font color="Olive" title="b4">f</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>  &amp; <font color="Olive" title="b4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be   <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font> &amp; <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 implies  ex <font color="Olive" title="b1">E</font> being   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 <a href="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a> ex <font color="Olive" title="b2">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="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">E</font></span>)</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> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></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="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> &amp; <font color="Olive" title="b2">f</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>  &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> ) )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:49"/><span class="lab"><font color="Green" title="E43">A1</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>
 <span class="kw">and </span><br/><a NAME="E2:49"/><span class="lab"><font color="Green" title="E44">A2</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b</font>
 <span class="kw">and </span><br/><a NAME="E3:49"/><span class="lab"><font color="Green" title="E45">A3</font></span>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">E</font> being   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 <a href="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a> ex <font color="Olive" title="b2">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="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">E</font></span>)</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> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></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="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> &amp; <font color="Olive" title="b2">f</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>  &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">B</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="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> <font color="Maroon" title="c1">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="ref" href="topreal1.html#T1" title="TOPREAL1:th.1">TOPREAL1:1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">F</font> being   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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>, <font color="Maroon" title="c9">s</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">F</font></span>)</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> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c7">B</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E6:49"/><span class="lab"><font color="Green" title="E46">A4</font></span>: 
<font color="Maroon" title="c8">F</font> <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"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">and </span><br/><a NAME="E7:49"/><span class="lab"><font color="Green" title="E47">A5</font></span>: 
<font color="Maroon" title="c9">s</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 <span class="kw">and </span><br/><a NAME="E8:49"/><span class="lab"><font color="Green" title="E48">A6</font></span>: 
<font color="Maroon" title="c9">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font>
 <span class="kw">and </span><br/><a NAME="E9:49"/><span class="lab"><font color="Green" title="E49">A7</font></span>: 
<font color="Maroon" title="c9">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="ref" href="borsuk_4.html#T43" target="_self" title="BORSUK_4:th.43">Th68</a></span>;<br/>
<span class="lab"><font color="Green" title="E50">A8</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">s</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">F</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:49"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c8">F</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>
;<br/>
<a NAME="E11:49"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A9</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="txt" href="#E6:49"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">E</font> = <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_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 <a href="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="txt" href="#E3:49"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="ref" href="borsuk_4.html#T36" target="_self" title="BORSUK_4:th.36">Th61</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">X</font> = <font color="Maroon" title="c10">E</font>;<br/>
<a NAME="E13:49"/><span class="lab"><font color="Green" title="E52">A10</font></span>: 
<a href="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c10">E</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of  <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<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">set </span><font color="Maroon" title="c12">sX</font> = <font color="Maroon" title="c9">s</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">E</font>;<br/>
<a NAME="E14:49"/><span class="lab"><font color="Green" title="E53">A11</font></span>: 
( <font color="Maroon" title="c9">s</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; <font color="Maroon" title="c9">s</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:49"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/>
<a NAME="E15:49"/><span class="lab"><font color="Green" title="E54">A12</font></span>: 
<font color="Maroon" title="c9">s</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:49"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/>
<a NAME="E16:49"/><span class="lab"><font color="Green" title="E55">A13</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> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T8" title="PRE_TOPC:th.8">PRE_TOPC:8</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">Ap</font> = <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <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> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<a NAME="E18:49"/>
 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> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T8" title="PRE_TOPC:th.8">PRE_TOPC:8</a></span>;<br/>
<a NAME="E19:49"/><span class="kw">then </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> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></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"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:49"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<a NAME="E20:49"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A14</font></span>: 
<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T4" title="TSEP_1:th.4">TSEP_1:4</a></span>;<br/>
<a NAME="E21:49"/><span class="lab"><font color="Green" title="E57">A15</font></span>: 
<font color="Maroon" title="c10">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:49"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="ref" href="xxreal_1.html#T23" title="XXREAL_1:th.23">XXREAL_1:23</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">X9</font> = <font color="Maroon" title="c10">E</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">F</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T8" title="PRE_TOPC:th.8">PRE_TOPC:8</a></span>;<br/>
<a NAME="E23:49"/><span class="lab"><font color="Green" title="E58">A16</font></span>: 
<span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c14">X9</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of  <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<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/>
<a NAME="E24:49"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c10">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T8" title="PRE_TOPC:th.8">PRE_TOPC:8</a></span>;<br/>
<a NAME="E25:49"/><span class="kw">then </span>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c10">E</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="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:49"><span class="lab"><font color="Green" title="E57">A15</font></span></a>, <a class="ref" href="pre_topc.html#T8" title="PRE_TOPC:th.8">PRE_TOPC:8</a></span>;<br/>
<a NAME="E26:49"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A17</font></span>: 
<a href="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c10">E</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:49"><span class="lab"><font color="Green" title="E52">A10</font></span></a>, <a class="ref" href="tsep_1.html#T4" title="TSEP_1:th.4">TSEP_1:4</a></span>;<br/>
<a NAME="E27:49"/><span class="lab"><font color="Green" title="E60">A18</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> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c13">Ap</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T7" title="PRE_TOPC:th.7">PRE_TOPC:7</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<span class="lab"><font color="Green" title="E61">A19</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">s</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">E</font></span>)</span> = 
<font color="Maroon" title="c10">E</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:49"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="txt" href="#E10:49"><span class="lab"><font color="Green" title="E50">A8</font></span></a>, <a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a>, <a class="ref" href="xxreal_1.html#T23" title="XXREAL_1:th.23">XXREAL_1:23</a></span>
<br/>.= 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c10">E</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>
;<br/>
<a NAME="E29:49"/><span class="lab"><font color="Green" title="E62">A20</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">s</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"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:49"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/>
<a NAME="E30:49"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A21</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E31:49"/>
<font color="Maroon" title="c10">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">F</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="txt" href="#E6:49"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="ref" href="xxreal_1.html#T134" title="XXREAL_1:th.134">XXREAL_1:134</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E64">A22</font></span>: <font color="Maroon" title="c9">s</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c10">E</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">s</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> (<font color="Maroon" title="c9">s</font>,<font color="Maroon" title="c5">a</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:49"><span class="lab"><font color="Green" title="E53">A11</font></span></a>, <a class="ref" href="funct_1.html#T64" title="FUNCT_1:th.64">FUNCT_1:64</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">s</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:49"><span class="lab"><font color="Green" title="E51">A9</font></span></a>, <a class="ref" href="funct_1.html#T59" title="FUNCT_1:th.59">FUNCT_1:59</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">s</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:49"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="txt" href="#E10:49"><span class="lab"><font color="Green" title="E50">A8</font></span></a>, <a class="ref" href="relat_1.html#T113" title="RELAT_1:th.113">RELAT_1:113</a></span>
<br/>.= 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</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> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:49"><span class="lab"><font color="Green" title="E63">A21</font></span></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>
;<br/>
<a NAME="E33:49"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A23</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</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> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">s</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">E</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T115" title="RELAT_1:th.115">RELAT_1:115</a></span>;<br/>
<a NAME="E34:49"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">s</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">E</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">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:49"><span class="lab"><font color="Green" title="E64">A22</font></span></a>, <a class="ref" href="relat_1.html#T115" title="RELAT_1:th.115">RELAT_1:115</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">sX</font> = <font color="Maroon" title="c9">s</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">E</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c10">E</font></span>)</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> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:49"><span class="lab"><font color="Green" title="E61">A19</font></span></a>, <a class="ref" href="funct_2.html#T1" title="FUNCT_2:th.1">FUNCT_2:1</a></span>;<br/>
<a NAME="E36:49"/><span class="lab"><font color="Green" title="E66">B1</font></span>: 
<font color="Maroon" title="c9">s</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:49"><span class="lab"><font color="Green" title="E62">A20</font></span></a>, <a class="txt" href="#E30:49"><span class="lab"><font color="Green" title="E63">A21</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E37:49"/><span class="lab"><font color="Green" title="E67">B2</font></span>: 
<font color="Maroon" title="c15">sX</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:49"><span class="lab"><font color="Green" title="E65">A23</font></span></a>, <a class="txt" href="#E30:49"><span class="lab"><font color="Green" title="E63">A21</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E38:49"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="ref" href="xxreal_1.html#T2" title="XXREAL_1:th.2">XXREAL_1:2</a></span>;<br/>
<a NAME="E39:49"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A24</font></span>: 
<font color="Maroon" title="c15">sX</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:49"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>;<br/>
<a NAME="E40:49"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c10">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T8" title="PRE_TOPC:th.8">PRE_TOPC:8</a></span>;<br/>
<a NAME="E41:49"/><span class="kw">then </span>
<a href="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c10">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c14">X9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:49"><span class="lab"><font color="Green" title="E52">A10</font></span></a>, <a class="txt" href="#E23:49"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="ref" href="pre_topc.html#T8" title="PRE_TOPC:th.8">PRE_TOPC:8</a>, <a class="ref" href="tsep_1.html#T5" title="TSEP_1:th.5">TSEP_1:5</a></span>;<br/>
<a NAME="E42:49"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A25</font></span>: 
<font color="Maroon" title="c15">sX</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="#E14:49"><span class="lab"><font color="Green" title="E53">A11</font></span></a>, <a class="txt" href="#E20:49"><span class="lab"><font color="Green" title="E56">A14</font></span></a>, <a class="ref" href="borsuk_4.html#T44" target="_self" title="BORSUK_4:th.44">Th69</a></span>;<br/>
<a NAME="E43:49"/><span class="lab"><font color="Green" title="E70">A26</font></span>: 
<font color="Maroon" title="c15">sX</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:49"><span class="lab"><font color="Green" title="E53">A11</font></span></a>, <a class="ref" href="funct_1.html#T52" title="FUNCT_1:th.52">FUNCT_1:52</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c15">sX</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  = 
<font color="Maroon" title="c15">sX</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:49"><span class="lab"><font color="Green" title="E67">B2</font></span></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">s</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">s</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c10">E</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:49"><span class="lab"><font color="Green" title="E53">A11</font></span></a>, <a class="ref" href="rfunct_2.html#T17" title="RFUNCT_2:th.17">RFUNCT_2:17</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">s</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</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">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:49"><span class="lab"><font color="Green" title="E66">B1</font></span></a>, <a class="txt" href="#E14:49"><span class="lab"><font color="Green" title="E53">A11</font></span></a>, <a class="txt" href="#E32:49"><span class="lab"><font color="Green" title="E64">A22</font></span></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">s</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">Ap</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>
;<br/>
<a NAME="E45:49"/><span class="kw">then </span>
<font color="Maroon" title="c15">sX</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:49"><span class="lab"><font color="Green" title="E54">A12</font></span></a>, <a class="txt" href="#E26:49"><span class="lab"><font color="Green" title="E59">A17</font></span></a>, <a class="txt" href="#E27:49"><span class="lab"><font color="Green" title="E60">A18</font></span></a>, <a class="ref" href="borsuk_4.html#T44" target="_self" title="BORSUK_4:th.44">Th69</a></span>;<br/>
<a NAME="E46:49"/><span class="kw">then </span>
<font color="Maroon" title="c15">sX</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:49"><span class="lab"><font color="Green" title="E65">A23</font></span></a>, <a class="txt" href="#E28:49"><span class="lab"><font color="Green" title="E61">A19</font></span></a>, <a class="txt" href="#E42:49"><span class="lab"><font color="Green" title="E69">A25</font></span></a>, <a class="txt" href="#E43:49"><span class="lab"><font color="Green" title="E70">A26</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E47:49"/>
 ex <font color="Olive" title="b1">E</font> being   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 <a href="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a> ex <font color="Olive" title="b2">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="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">E</font></span>)</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> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></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="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> &amp; <font color="Olive" title="b2">f</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>  &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:49"><span class="lab"><font color="Green" title="E68">A24</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
