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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</font>, <font color="Olive" title="b2">N</font> being   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>  st <font color="Olive" title="b1">M</font> is <font color="Maroon" title="c1">n</font> <a href="mfold_1.html#V2" title="MFOLD_1:attr.2">-locally_euclidean</a>  &amp; <font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">N</font> <a href="mfold_2.html#R1" title="MFOLD_2:pred.1">are_homeomorphic</a>  holds <br/><font color="Olive" title="b2">N</font> is <font color="Maroon" title="c1">n</font> <a href="mfold_1.html#V2" title="MFOLD_1:attr.2">-locally_euclidean</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">M</font>, <font color="Maroon" title="c3">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"> ( <font color="Maroon" title="c2">M</font> is <font color="Maroon" title="c1">n</font> <a href="mfold_1.html#V2" title="MFOLD_1:attr.2">-locally_euclidean</a>  &amp; <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c3">N</font> <a href="mfold_2.html#R1" title="MFOLD_2:pred.1">are_homeomorphic</a>  implies <font color="Maroon" title="c3">N</font> is <font color="Maroon" title="c1">n</font> <a href="mfold_1.html#V2" title="MFOLD_1:attr.2">-locally_euclidean</a>  )</span><br/>





<span class="kw">assume </span><a NAME="E1:13"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
<font color="Maroon" title="c2">M</font> is <font color="Maroon" title="c1">n</font> <a href="mfold_1.html#V2" title="MFOLD_1:attr.2">-locally_euclidean</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c3">N</font> <a href="mfold_2.html#R1" title="MFOLD_2:pred.1">are_homeomorphic</a>  or <font color="Maroon" title="c3">N</font> is <font color="Maroon" title="c1">n</font> <a href="mfold_1.html#V2" title="MFOLD_1:attr.2">-locally_euclidean</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:13"/>
<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c3">N</font> <a href="mfold_2.html#R1" title="MFOLD_2: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"> <font color="Maroon" title="c3">N</font> is <font color="Maroon" title="c1">n</font> <a href="mfold_1.html#V2" title="MFOLD_1:attr.2">-locally_euclidean</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c2">M</font><span class="kw"> such that </span><br/><a NAME="E4:13"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
<font color="Maroon" title="c4">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/>
<a NAME="E5:13"/><span class="lab"><font color="Green" title="E13">A3</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c3">N</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c4">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c4">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; <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>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:13"><span class="lab"><font color="Green" title="E12">A2</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="E6:13"/>
 for <font color="Olive" title="b1">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">N</font>  ex <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">q</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 <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> st <font color="Olive" title="b2">U</font>,<font color="Olive" title="b3">S</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">N</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">U</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c5">q</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 <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> st <font color="Olive" title="b1">U</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">set </span><font color="Maroon" title="c6">p</font> = <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">q</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">U1</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">q</font>, <font color="Maroon" title="c8">S1</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="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E2:13_1"/><span class="lab"><font color="Green" title="E14">A4</font></span>: 
<font color="Maroon" title="c7">U1</font>,<font color="Maroon" title="c8">S1</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="#E1:13"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="ref" href="mfold_1.html#D4" title="MFOLD_1:def.4">MFOLD_1:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">U2</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">M</font>, <font color="Maroon" title="c10">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 <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"> such that </span><br/><a NAME="E4:13_1"/><span class="lab"><font color="Green" title="E15">A5</font></span>: 
( <font color="Maroon" title="c9">U2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">U1</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">U2</font> &amp; <font color="Maroon" title="c9">U2</font>,<font color="Maroon" title="c10">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="#E2:13_1"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="mfold_1.html#T11" title="MFOLD_1:th.11">MFOLD_1:11</a></span>;<br/>
<a NAME="E5:13_1"/><span class="lab"><font color="Green" title="E16">A6</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c9">U2</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="#E5:13"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="tops_2.html#T43" title="TOPS_2:th.43">TOPS_2:43</a></span>;<br/>
<a NAME="E6:13_1"/>
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c9">U2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:13_1"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="ref" href="funct_2.html#T38" title="FUNCT_2:th.38">FUNCT_2:38</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">U</font> = <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c9">U2</font> as    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c5">q</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:13_1"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="ref" href="connsp_2.html#T6" title="CONNSP_2:th.6">CONNSP_2:6</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c11">U</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 <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> st <font color="Maroon" title="c11">U</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="c10">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="c11">U</font>,<font color="Maroon" title="c10">S</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a> </span><br/>

<a NAME="E8:13_1"/>
<font color="Maroon" title="c11">U</font>,<font color="Maroon" title="c9">U2</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="#E4:13"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="mfold_2.html#T5" target="_self" title="MFOLD_2:th.5">Th5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:13_1"/>
<font color="Maroon" title="c11">U</font>,<font color="Maroon" title="c10">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="#E4:13_1"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="ref" href="mfold_2.html#T8" target="_self" title="MFOLD_2:th.8">Th8</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><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:13"/>
<font color="Maroon" title="c3">N</font> is <font color="Maroon" title="c1">n</font> <a href="mfold_1.html#V2" title="MFOLD_1:attr.2">-locally_euclidean</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mfold_1.html#D4" title="MFOLD_1:def.4">MFOLD_1:def 4</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>
