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<b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">r</font>, <font color="Maroon" title="c5">s</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">s</font> implies for <font color="Olive" title="b1">h</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<span class="p1">(<span class="default"><a href="topreala.html#K1" title="TOPREALA:func.1">Trectangle</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font></span>)</span>  st <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> holds <br/><font color="Olive" title="b1">h</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  )</span><br/>







<b>assume </b><a NAME="E1:53"/><i><font color="Green" title="E23">A1</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">s</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">h</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<span class="p1">(<span class="default"><a href="topreala.html#K1" title="TOPREALA:func.1">Trectangle</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font></span>)</span>  st <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> holds <br/><font color="Olive" title="b1">h</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> </span><br/>

<b>set </b><font color="Maroon" title="c6">C1</font> =  <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>;<br/>
<b>set </b><font color="Maroon" title="c7">C2</font> =  <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font>;<br/>
<b>set </b><font color="Maroon" title="c8">TR</font> =  <a href="topreala.html#K1" title="TOPREALA:func.1">Trectangle</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font>;<br/>
<b>let </b><font color="Maroon" title="c9">h</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<span class="p1">(<span class="default"><a href="topreala.html#K1" title="TOPREALA:func.1">Trectangle</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> implies <font color="Maroon" title="c9">h</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  )</span><br/>

<b>assume </b><a NAME="E2:53"/><i><font color="Green" title="E24">A2</font></i>: 
<font color="Maroon" title="c9">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">h</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> </span><br/>

<b>reconsider </b><font color="Maroon" title="c10">S0</font> = <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <b>by </b><i><a class="ref" href="borsuk_3.html#T25" title="BORSUK_3:th.25">BORSUK_3:25</a></i>;<br/>
<a NAME="E4:53"/><i><font color="Green" title="E25">A3</font></i>: 
 <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">s</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="jgraph_6.html#D2" title="JGRAPH_6:def.2">JGRAPH_6:def 2</a></i>;<br/>
<a NAME="E5:53"/><i><font color="Green" title="E26">A4</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topreala.html#K1" title="TOPREALA:func.1">Trectangle</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<b>set </b><font color="Maroon" title="c11">p</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">r</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>;<br/>
<a NAME="E6:53"/>
( <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">r</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">r</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E7:53"/><b>then </b>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">r</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font>
 
<b>by </b><i><a class="txt" href="topreala.html#E1:53"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="topreala.html#E4:53"><i><font color="Green" title="E25">A3</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">T0</font> =  <a href="topreala.html#K1" title="TOPREALA:func.1">Trectangle</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2 <b>by </b><i><a class="txt" href="topreala.html#E5:53"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">g</font> = <font color="Maroon" title="c9">h</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c10">S0</font>,<font color="Maroon" title="c12">T0</font> ;<br/>
<a NAME="E10:53"/><i><font color="Green" title="E27">A5</font></i>: 
<font color="Maroon" title="c13">g</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <font color="Maroon" title="c10">S0</font>
 
<b>by </b><i><a class="txt" href="topreala.html#E2:53"><i><font color="Green" title="E24">A2</font></i></a></i>;<br/>
<a NAME="E11:53"/>
<font color="Maroon" title="c13">g</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:53_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">g</font> <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 <font color="Maroon" title="c12">T0</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>,<a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">T0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">g</font></span><br/>

<b>let </b><font color="Maroon" title="c14">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">T0</font> or <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">g</font> )</span><br/>

<b>assume </b><a NAME="E2:53_1"/>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">T0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">g</font></span><br/>

<b>then consider </b><font color="Maroon" title="c15">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:53_1"/><i><font color="Green" title="E28">A6</font></i>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">p</font>
 <b>and </b><br/><a NAME="E5:53_1"/><i><font color="Green" title="E29">A7</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c15">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c15">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">s</font> )
 <b>by </b><i><a class="txt" href="topreala.html#E4:53"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="topreala.html#E5:53"><i><font color="Green" title="E26">A4</font></i></a></i>;<br/>
<a NAME="E6:53_1"/><i><font color="Green" title="E30">A8</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>,<span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topreala.html#E1:53"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="topreala.html#T48" target="_self" title="TOPREALA:th.48">Th48</a></i>;<br/>
<a NAME="E7:53_1"/><i><font color="Green" title="E31">A9</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">g</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 <font color="Maroon" title="c10">S0</font>
 
<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="E8:53_1"/>
( <font color="Maroon" title="c15">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c15">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 
<b>by </b><i><a class="txt" href="topreala.html#E5:53_1"><i><font color="Green" title="E29">A7</font></i></a></i>;<br/>
<a NAME="E9:53_1"/><b>then </b><i><font color="Green" title="E32">A10</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">g</font>
 
<b>by </b><i><a class="txt" href="topreala.html#E6:53_1"><i><font color="Green" title="E30">A8</font></i></a>, <a class="txt" href="topreala.html#E7:53_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>then </b><font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="topreala.html#E2:53"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
<b>by </b><i><a class="ref" href="topreala.html#D2" target="_self" title="TOPREALA:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c14">y</font>
<b>by </b><i><a class="txt" href="topreala.html#E4:53_1"><i><font color="Green" title="E28">A6</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>
;<br/>
<b>hence </b><a NAME="E11:53_1"/>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">g</font>
 <b>by </b><i><a class="txt" href="topreala.html#E9:53_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:53"/>
<font color="Maroon" title="c9">h</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 <b>by </b><i><a class="txt" href="topreala.html#E10:53"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="topreala.html#T56" target="_self" title="TOPREALA:th.56">Th56</a>, <a class="ref" href="jordan16.html#T13" title="JORDAN16:th.13">JORDAN16:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
