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<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c2">Y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>,<font color="Maroon" title="c2">Y</font>;<br/>





<b>set </b><font color="Maroon" title="c5">Z</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<b>assume </b><a NAME="E1:21"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>

<a NAME="E2:21"/><i><font color="Green" title="E12">A2</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<b>thus </b> <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c2">Y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c2">Y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
<b>by </b><i><a class="ref" href="pre_topc.html#T12" title="PRE_TOPC:th.12">PRE_TOPC:12</a></i>
; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a><br/>

<b>thus </b> <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font>
<b>by </b><i><a class="txt" href="borsuk_3.html#E1:21"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="funct_3.html#T60" title="FUNCT_3:th.60">FUNCT_3:60</a></i>
<br/>.= 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">Y</font>
<b>by </b><i><a class="ref" href="pre_topc.html#T12" title="PRE_TOPC:th.12">PRE_TOPC:12</a></i>
;<br/>

<b>thus </b><a NAME="E5:21"/>
<font color="Maroon" title="c4">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="borsuk_3.html#E1:21"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="borsuk_3.html#T6" target="_self" title="BORSUK_3:th.6">Th6</a></i>;<br/>

<b>thus </b><a NAME="E6:21"/>
<font color="Maroon" title="c4">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="borsuk_3.html#E1:21"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="borsuk_3.html#E2:21"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="yellow12.html#T39" title="YELLOW12:th.39">YELLOW12:39</a></i>;<br/>

<b>reconsider </b><font color="Maroon" title="c6">Z</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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 <font color="Maroon" title="c1">X</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">idZ</font> = <font color="Maroon" title="c2">Y</font> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c3">x</font> as  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">Y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">Z</font></span>)</span> <b>by </b><i><a class="ref" href="borsuk_3.html#T2" target="_self" title="BORSUK_3:th.2">Th2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">KA</font> = <span class="p1"><a href="funct_3.html#K14" title="FUNCT_3:func.14">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c7">idZ</font></span><a href="funct_3.html#K14" title="FUNCT_3:func.14">:&gt;</a></span> as  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">Y</font>,<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c2">Y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">Z</font></span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <b>by </b><i><a class="ref" href="yellow12.html#T41" title="YELLOW12:th.41">YELLOW12:41</a></i>;<br/>
<a NAME="E10:21"/>
<font color="Maroon" title="c8">KA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> 
 
<b>by </b><i><a class="txt" href="borsuk_3.html#E1:21"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="borsuk_3.html#T8" target="_self" title="BORSUK_3:th.8">Th8</a></i>;<br/>
<b>hence </b><a NAME="E11:21"/>
<font color="Maroon" title="c4">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 ;<br/>


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