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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">A</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<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><br/> for <font color="Olive" title="b2">B</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<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> holds   <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default">1,2 <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span>)</span> is   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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> )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c6">T</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/>
<span class="kw">set </span><font color="Maroon" title="c7">S</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/>
<span class="kw">assume </span><a NAME="E1:71"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
( <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<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><br/> for <font color="Olive" title="b2">B</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<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> holds   <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default">1,2 <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span>)</span> is   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">h</font> = <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#K3" title="BORSUK_1:func.3">[:</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#K3" title="BORSUK_1:func.3">:]</a></span> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</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#K3" title="BORSUK_1:func.3">:]</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> <span class="kw">by</span> <span class="lab"><a class="ref" href="topreala.html#T57" target="_self" title="TOPREALA:th.57">Th57</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">A</font> be   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">B</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<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> holds   <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default">1,2 <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c9">A</font>,<font color="Olive" title="b1">B</font></span>)</span> is   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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></span><br/><span class="kw">let </span><font color="Maroon" title="c10">B</font> be   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default">1,2 <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c9">A</font>,<font color="Maroon" title="c10">B</font></span>)</span> is   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c11">P</font> =  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default">1,2 <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c9">A</font>,<font color="Maroon" title="c10">B</font></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="topreala.html#T60" target="_self" title="TOPREALA:th.60">Th60</a></span>;<br/>
<a NAME="E4:71"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c9">A</font>,<font color="Maroon" title="c10">B</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T46" title="BORSUK_1:th.46">BORSUK_1:46</a></span>;<br/>
<a NAME="E5:71"/>
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> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font></span>)</span> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T1" title="TSEP_1:th.1">TSEP_1:1</a></span>;<br/>
<a NAME="E6:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A3</font></span>: 
<font color="Maroon" title="c10">B</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E7:71"/>
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> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T1" title="TSEP_1:th.1">TSEP_1:1</a></span>;<br/>
<a NAME="E8:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A4</font></span>: 
<font color="Maroon" title="c9">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="lab"><font color="Green" title="E28">A5</font></span>: <font color="Maroon" title="c8">h</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c9">A</font>,<font color="Maroon" title="c10">B</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> = 
<a href="topreala.html#K2" title="TOPREALA:func.2">R2Homeomorphism</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c9">A</font>,<font color="Maroon" title="c10">B</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T162" title="RELAT_1:th.162">RELAT_1:162</a></span>
<br/>.= 
<font color="Maroon" title="c11">P</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:71"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E6:71"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="topreala.html#T55" target="_self" title="TOPREALA:th.55">Th55</a></span>
;<br/>
<a NAME="E10:71"/>
( <font color="Maroon" title="c8">h</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>  &amp; not  <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> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="topreala.html#T54" target="_self" title="TOPREALA:th.54">Th54</a>, <a class="ref" href="topreala.html#T58" target="_self" title="TOPREALA:th.58">Th58</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:71"/>
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default">1,2 <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c9">A</font>,<font color="Maroon" title="c10">B</font></span>)</span> is   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:71"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E4:71"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="topgrp_1.html#T25" title="TOPGRP_1:th.25">TOPGRP_1:25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
