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<span class="kw">let </span><font color="Maroon" title="c2">M</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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_0.html#V2" title="MFOLD_0:attr.2">-locally_euclidean</a>  implies <font color="Maroon" title="c2">M</font> is  <a href="struct_0.html#NV10" title="STRUCT_0:NV.10">infinite</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">TR</font> =  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font>;<br/>
<span class="kw">assume </span><a NAME="E1:4_2_1"/><span class="lab"><font color="Green" title="E3">A8</font></span>: 
<font color="Maroon" title="c2">M</font> is <font color="Maroon" title="c1">n</font> <a href="mfold_0.html#V2" title="MFOLD_0:attr.2">-locally_euclidean</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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  <a href="struct_0.html#NV10" title="STRUCT_0:NV.10">infinite</a> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:4_2_1"/><span class="lab"><font color="Green" title="E4">A9</font></span>: 
<font color="Maroon" title="c4">p</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="c2">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">p</font> = <font color="Maroon" title="c4">p</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:4_2_1"><span class="lab"><font color="Green" title="E4">A9</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">U</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c5">p</font><span class="kw"> such that </span><br/><a NAME="E6:4_2_1"/><span class="lab"><font color="Green" title="E5">A10</font></span>: 
<font color="Maroon" title="c2">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">U</font>, <a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<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>)</span>,1) <a href="mfold_2.html#R1" title="MFOLD_2:pred.1">are_homeomorphic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:4_2_1"><span class="lab"><font color="Green" title="E3">A8</font></span></a>, <a class="ref" href="mfold_0.html#D3" onmouseover="rs('mfold_0/D3')" onmouseout="rh()">MFOLD_0:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<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>)</span>,1)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">U</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E8:4_2_1"/><span class="lab"><font color="Green" title="E6">A11</font></span>: 
<font color="Maroon" title="c7">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="txt" href="#E6:4_2_1"><span class="lab"><font color="Green" title="E5">A10</font></span></a>, <a class="ref" href="t_0topsp.html#D1" onmouseover="rs('t_0topsp/D1')" onmouseout="rh()">T_0TOPSP:def 1</a></span>;<br/>
<a NAME="E9:4_2_1"/><span class="lab"><font color="Green" title="E7">C</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<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>)</span>,1)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<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>)</span>,1)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E10:4_2_1"/><span class="lab"><font color="Green" title="E8">A12</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<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>)</span>,1)</span>)</span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">U</font></span>)</span> &amp; <font color="Maroon" title="c7">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:4_2_1"><span class="lab"><font color="Green" title="E6">A11</font></span></a>, <a class="ref" href="tops_2.html#T58" onmouseover="rs('tops_2/T58')" onmouseout="rh()">TOPS_2:58</a></span>;<br/>
<a NAME="E11:4_2_1"/>
(  not  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<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>)</span>,1) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<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>)</span>,1) is  <a href="mfold_1.html#V1" title="MFOLD_1:attr.1">ball</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mfold_1.html#D1" onmouseover="rs('mfold_1/D1')" onmouseout="rh()">MFOLD_1:def 1</a></span>;<br/>
<a NAME="E12:4_2_1"/><span class="kw">then </span>
<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> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<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>)</span>,1)</span>)</span>,<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> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<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>)</span> <a href="mfold_2.html#R1" title="MFOLD_2:pred.1">are_homeomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mfold_1.html#T10" onmouseover="rs('mfold_1/T10')" onmouseout="rh()">MFOLD_1:10</a>, <a class="ref" href="metrizts.html#D1" onmouseover="rs('metrizts/D1')" onmouseout="rh()">METRIZTS:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<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>)</span>,1)</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<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>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E14:4_2_1"/><span class="lab"><font color="Green" title="E9">B11</font></span>: 
<font color="Maroon" title="c8">g</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 ;<br/>
<a NAME="E15:4_2_1"/><span class="lab"><font color="Green" title="E10">B12</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<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>)</span>,1)</span>)</span></span>)</span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<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>)</span></span>)</span> &amp; <font color="Maroon" title="c8">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:4_2_1"><span class="lab"><font color="Green" title="E9">B11</font></span></a>, <a class="ref" href="tops_2.html#T58" onmouseover="rs('tops_2/T58')" onmouseout="rh()">TOPS_2:58</a></span>;<br/>
<a NAME="E16:4_2_1"/>
(  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<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>)</span>,1) is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>  &amp;  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<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>)</span>,1) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<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>)</span>,1) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:4_2_1"><span class="lab"><font color="Green" title="E10">B12</font></span></a>, <a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a>, <a class="ref" href="topreal9.html#T16" onmouseover="rs('topreal9/T16')" onmouseout="rh()">TOPREAL9:16</a></span>;<br/>
<a NAME="E17:4_2_1"/><span class="kw">then </span>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<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>)</span>,1)</span>)</span> is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:4_2_1"><span class="lab"><font color="Green" title="E7">C</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:4_2_1"/>
<font color="Maroon" title="c2">M</font> is  <a href="struct_0.html#NV10" title="STRUCT_0:NV.10">infinite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:4_2_1"><span class="lab"><font color="Green" title="E8">A12</font></span></a>, <a class="ref" href="card_1.html#T59" onmouseover="rs('card_1/T59')" onmouseout="rh()">CARD_1:59</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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