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<span class="kw">let </span><font color="Maroon" title="c1">M</font>, <font color="Maroon" title="c2">N</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</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">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><br/>  for <font color="Olive" title="b2">U</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Olive" title="b1">p</font><br/>  for <font color="Olive" title="b3">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 <font color="Maroon" title="c2">N</font>  st <font color="Olive" title="b2">U</font>,<font color="Olive" title="b3">B</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a>  holds <br/> ex <font color="Olive" title="b4">V</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 <font color="Maroon" title="c1">M</font> ex <font color="Olive" title="b5">S</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 <font color="Maroon" title="c2">N</font> st <br/>( <font color="Olive" title="b4">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">U</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">V</font> &amp; <font color="Olive" title="b4">V</font>,<font color="Olive" title="b5">S</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>; <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">U</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">p</font><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 <font color="Maroon" title="c2">N</font>  st <font color="Olive" title="b1">U</font>,<font color="Olive" title="b2">B</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a>  holds <br/> ex <font color="Olive" title="b3">V</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 <font color="Maroon" title="c1">M</font> ex <font color="Olive" title="b4">S</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 <font color="Maroon" title="c2">N</font> st <br/>( <font color="Olive" title="b3">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">U</font> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">V</font> &amp; <font color="Olive" title="b3">V</font>,<font color="Olive" title="b4">S</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">U</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">p</font>; <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">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 <font color="Maroon" title="c2">N</font>  st <font color="Maroon" title="c4">U</font>,<font color="Olive" title="b1">B</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a>  holds <br/> ex <font color="Olive" title="b2">V</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 <font color="Maroon" title="c1">M</font> ex <font color="Olive" title="b3">S</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 <font color="Maroon" title="c2">N</font> st <br/>( <font color="Olive" title="b2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U</font> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">V</font> &amp; <font color="Olive" title="b2">V</font>,<font color="Olive" title="b3">S</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">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 <font color="Maroon" title="c2">N</font>; <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="c4">U</font>,<font color="Maroon" title="c5">B</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a>  implies  ex <font color="Olive" title="b1">V</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 <font color="Maroon" title="c1">M</font> ex <font color="Olive" title="b2">S</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 <font color="Maroon" title="c2">N</font> st <br/>( <font color="Olive" title="b1">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U</font> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">S</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a>  ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:18"/>
<font color="Maroon" title="c4">U</font>,<font color="Maroon" title="c5">B</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a> 
 ; <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">V</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 <font color="Maroon" title="c1">M</font> ex <font color="Olive" title="b2">S</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 <font color="Maroon" title="c2">N</font> st <br/>( <font color="Olive" title="b1">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U</font> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">S</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a>  )</span><br/>

<a NAME="E2:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A1</font></span>: 
<font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">U</font>,<font color="Maroon" title="c2">N</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">B</font> <a href="borsuk_3.html#R1" title="BORSUK_3:pred.1">are_homeomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metrizts.html#D1" title="METRIZTS:def.1">METRIZTS:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">U</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">B</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:18"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
<font color="Maroon" title="c6">f</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="ref" href="t_0topsp.html#D1" title="T_0TOPSP:def.1">T_0TOPSP:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font><span class="kw"> such that </span><br/><a NAME="E6:18"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
( <font color="Maroon" title="c7">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c7">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U</font> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_2.html#T6" title="CONNSP_2:th.6">CONNSP_2:6</a></span>;<br/>
<a NAME="E7:18"/>
<font color="Maroon" title="c7">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">U</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:18"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">V1</font> = <font color="Maroon" title="c7">V</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">U</font></span>)</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">M1</font> = <font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">U</font> as   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:18"><span class="lab"><font color="Green" title="E15">A3</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">N1</font> = <font color="Maroon" title="c2">N</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">B</font> as   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:18"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="yellow14.html#T18" title="YELLOW14:th.18">YELLOW14:18</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">f1</font> = <font color="Maroon" title="c6">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c9">M1</font>,<font color="Maroon" title="c10">N1</font> ;<br/>
<a NAME="E12:18"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">B</font></span>)</span>
 
;<br/>
<a NAME="E13:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A4</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</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="E14:18"/>
<font color="Maroon" title="c8">V1</font>,<font color="Maroon" title="c6">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">V1</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="metrizts.html#T3" title="METRIZTS:th.3">METRIZTS:3</a></span>;<br/>
<a NAME="E15:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A5</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">U</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">V1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">B</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="c6">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">V1</font></span>)</span> <a href="borsuk_3.html#R1" title="BORSUK_3:pred.1">are_homeomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metrizts.html#D1" title="METRIZTS:def.1">METRIZTS:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">V</font> = <font color="Maroon" title="c7">V</font> as   <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 <font color="Maroon" title="c1">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:18"><span class="lab"><font color="Green" title="E15">A3</font></span></a></span>;<br/>
<a NAME="E17:18"/>
<font color="Maroon" title="c5">B</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 <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">B</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/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">N1</font> = <font color="Maroon" title="c2">N</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">B</font> as   <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c2">N</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T16" title="TSEP_1:th.16">TSEP_1:16</a></span>;<br/>
<a NAME="E19:18"/>
<font color="Maroon" title="c5">B</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="c2">N</font>
 
;<br/>
<a NAME="E20:18"/><span class="kw">then </span>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">B</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 <font color="Maroon" title="c2">N</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/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">S</font> = <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">V1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">N</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">S1</font> = <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">V1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c13">N1</font> ;<br/>
<a NAME="E23:18"/><span class="lab"><font color="Green" title="E18">A6</font></span>: 
 for <font color="Olive" title="b1">P</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c9">M1</font> holds <br/> ( <font color="Olive" title="b1">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  iff <font color="Maroon" title="c11">f1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="topgrp_1.html#T25" title="TOPGRP_1:th.25">TOPGRP_1:25</a></span>;<br/>
<a NAME="E24:18"/><span class="lab"><font color="Green" title="E19">A7</font></span>: 
<font color="Maroon" title="c12">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D2" title="PRE_TOPC:def.2">PRE_TOPC:def 2</a></span>;<br/>
<a NAME="E25:18"/>
<font color="Maroon" title="c8">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">V</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c9">M1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<a NAME="E26:18"/><span class="kw">then </span>
<font color="Maroon" title="c8">V1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c9">M1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:18"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="ref" href="pre_topc.html#D4" title="PRE_TOPC:def.4">PRE_TOPC:def 4</a></span>;<br/>
<a NAME="E27:18"/><span class="kw">then </span>
<font color="Maroon" title="c8">V1</font> 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="pre_topc.html#D2" title="PRE_TOPC:def.2">PRE_TOPC:def 2</a></span>;<br/>
<a NAME="E28:18"/><span class="kw">then </span>
<font color="Maroon" title="c15">S1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:18"><span class="lab"><font color="Green" title="E18">A6</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">S</font> = <font color="Maroon" title="c14">S</font> as   <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 <font color="Maroon" title="c2">N</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T17" title="TSEP_1:th.17">TSEP_1:17</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c12">V</font>
; <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">S</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 <font color="Maroon" title="c2">N</font> st <br/>( <font color="Maroon" title="c12">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U</font> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">V</font> &amp; <font color="Maroon" title="c12">V</font>,<font color="Olive" title="b1">S</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a>  )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c16">S</font>
; <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="c12">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U</font> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">V</font> &amp; <font color="Maroon" title="c12">V</font>,<font color="Maroon" title="c16">S</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E30:18"/>
( <font color="Maroon" title="c12">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U</font> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">V</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:18"><span class="lab"><font color="Green" title="E15">A3</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"> <font color="Maroon" title="c12">V</font>,<font color="Maroon" title="c16">S</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a> </span><br/>

<a NAME="E31:18"/><span class="lab"><font color="Green" title="E20">A8</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">U</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c12">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:18"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="ref" href="pre_topc.html#T7" title="PRE_TOPC:th.7">PRE_TOPC:7</a></span>;<br/>
<a NAME="E32:18"/>
<font color="Maroon" title="c6">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T111" title="RELAT_1:th.111">RELAT_1:111</a></span>;<br/>
<a NAME="E33:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A9</font></span>: 
<font color="Maroon" title="c6">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:18"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E34:18"/>
<font color="Maroon" title="c6">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:18"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="ref" href="relat_1.html#T123" title="RELAT_1:th.123">RELAT_1:123</a></span>;<br/>
<a NAME="E35:18"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">B</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="c6">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">V1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">N</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c16">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:18"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="ref" href="pre_topc.html#T7" title="PRE_TOPC:th.7">PRE_TOPC:7</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E36:18"/>
<font color="Maroon" title="c12">V</font>,<font color="Maroon" title="c16">S</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:18"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="#E31:18"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="ref" href="metrizts.html#D1" title="METRIZTS:def.1">METRIZTS:def 1</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>
