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<b>let </b><font color="Maroon" title="c1">P</font>, <font color="Maroon" title="c2">Q</font> be   <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> 2</span>)</span>;<br/><b>let </b><font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c4">p2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>



<b>assume </b><a NAME="E1:28"/><i><font color="Green" title="E21">A1</font></i>: 
( <font color="Maroon" title="c1">P</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">Q</font> &amp; <font color="Maroon" title="c1">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Q</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c1">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> )
 ;<br/>

<a NAME="E2:28"/><b>then </b>
<font color="Maroon" title="c1">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:28"/><i><font color="Green" title="E22">A2</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Q</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">P</font> = <font color="Maroon" title="c1">P</font> 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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan5a.html#E4:28"><i><font color="Green" title="E22">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</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> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">P</font></span>)</span><b> such that </b><br/><a NAME="E7:28"/><i><font color="Green" title="E23">A3</font></i>: 
( <font color="Maroon" title="c7">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p1</font> &amp; <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p2</font> )
 <b>by </b><i><a class="txt" href="jordan5a.html#E1:28"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="topreal1.html#D2" title="TOPREAL1:def.2">TOPREAL1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">PP</font> = <font color="Maroon" title="c6">P</font> as   <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> 2</span>)</span> ;<br/>
<a NAME="E9:28"/><i><font color="Green" title="E24">A4</font></i>: 
<font color="Maroon" title="c8">PP</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E1:28"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="jordan5a.html#T1" target="_self" title="JORDAN5A:th.1">Th1</a></i>;<br/>
<a NAME="E10:28"/><i><font color="Green" title="E25">A5</font></i>: 
( <font color="Maroon" title="c6">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c6">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Q</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E11:28"/><b>then </b><i><font color="Green" title="E26">A6</font></i>: 
<font color="Maroon" title="c6">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Q</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E1:28"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="jordan5a.html#E9:28"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="compts_1.html#T18" title="COMPTS_1:th.18">COMPTS_1:18</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">g</font> = <font color="Maroon" title="c7">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> ;<br/>
<a NAME="E12:28"/><i><font color="Green" title="E27">A7</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 
<b>by </b><i><a class="txt" href="jordan5a.html#E7:28"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="tops_2.html#T70" title="TOPS_2:th.70">TOPS_2:70</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">f1</font> = <font color="Maroon" title="c7">f</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<a NAME="E14:28"/><i><font color="Green" title="E28">A8</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f</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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<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 NAME="E15:28"/><i><font color="Green" title="E29">A9</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">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> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">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"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">P</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> )
 
<b>by </b><i><a class="txt" href="jordan5a.html#E7:28"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E16:28"/><i><font color="Green" title="E30">A10</font></i>: 
<font color="Maroon" title="c7">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>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E7:28"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E17:28"/>
 <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> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">PQ</font> = <font color="Maroon" title="c6">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Q</font> 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 <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">P</font></span>)</span> <b>by </b><i><a class="txt" href="jordan5a.html#E1:28"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="jordan5a.html#E10:28"><i><font color="Green" title="E25">A5</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E19:28"/><i><font color="Green" title="E31">A11</font></i>: 
( <font color="Maroon" title="c6">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> &amp; <font color="Maroon" title="c11">PQ</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</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> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">P</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan5a.html#E1:28"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">P1</font> = <font color="Maroon" title="c6">P</font>, <font color="Maroon" title="c13">Q1</font> = <font color="Maroon" title="c2">Q</font> 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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan5a.html#E1:28"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">PQ1</font> = <font color="Maroon" title="c6">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Q</font> 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 <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c12">P1</font></span>)</span> <b>by </b><i><a class="txt" href="jordan5a.html#E19:28"><i><font color="Green" title="E31">A11</font></i></a></i>;<br/>
<a NAME="E22:28"/><i><font color="Green" title="E32">A12</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">P1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">Q1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c12">P1</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c14">PQ1</font>
 
<b>by </b><i><a class="ref" href="goboard9.html#T4" title="GOBOARD9:th.4">GOBOARD9:4</a></i>;<br/>
<a NAME="E23:28"/>
<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</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">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Q</font></span>)</span> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E11:28"><i><font color="Green" title="E26">A6</font></i></a>, <a class="txt" href="jordan5a.html#E19:28"><i><font color="Green" title="E31">A11</font></i></a>, <a class="ref" href="compts_1.html#T12" title="COMPTS_1:th.12">COMPTS_1:12</a></i>;<br/>
<a NAME="E24:28"/><b>then </b><i><font color="Green" title="E33">A13</font></i>: 
<font color="Maroon" title="c11">PQ</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E22:28"><i><font color="Green" title="E32">A12</font></i></a>, <a class="ref" href="compts_1.html#T12" title="COMPTS_1:th.12">COMPTS_1:12</a></i>;<br/>
<a NAME="E25:28"/><i><font color="Green" title="E34">A14</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#D9" title="PRE_TOPC:def.9">PRE_TOPC:def 9</a></i>;<br/>
<a NAME="E26:28"/><i><font color="Green" title="E35">A15</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c11">PQ</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
;<br/>
<a NAME="E27:28"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c11">PQ</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> 
 
<b>by </b><i><a class="txt" href="jordan5a.html#E25:28"><i><font color="Green" title="E34">A14</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">GPQ</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c11">PQ</font> as   <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>  ;<br/>
<a NAME="E29:28"/>
for <font color="Olive" title="b1">P</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>   st <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">GPQ</font> holds <br/><font color="Olive" title="b1">P</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E16:28"><i><font color="Green" title="E30">A10</font></i></a>, <a class="txt" href="jordan5a.html#E24:28"><i><font color="Green" title="E33">A13</font></i></a>, <a class="ref" href="weierstr.html#T14" title="WEIERSTR:th.14">WEIERSTR:14</a></i>;<br/>
<a NAME="E30:28"/><b>then </b>
<font color="Maroon" title="c15">GPQ</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E26:28"><i><font color="Green" title="E35">A15</font></i></a>, <a class="ref" href="compts_1.html#T11" title="COMPTS_1:th.11">COMPTS_1:11</a></i>;<br/>
<a NAME="E31:28"/><b>then </b><i><font color="Green" title="E36">A16</font></i>: 
(  <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c15">GPQ</font> is <a href="seq_4.html#V3" title="SEQ_4:attr.3">bounded</a> &amp;  <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c15">GPQ</font> is <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a> )
 
<b>by </b><i><a class="ref" href="weierstr.html#T17" title="WEIERSTR:th.17">WEIERSTR:17</a>, <a class="ref" href="weierstr.html#T18" title="WEIERSTR:th.18">WEIERSTR:18</a></i>;<br/>
<a NAME="E32:28"/><b>then </b><i><font color="Green" title="E37">A17</font></i>: 
 <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c15">GPQ</font> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#D3" title="SEQ_4:def.3">SEQ_4:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">OO</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E34:28"/><i><font color="Green" title="E38">A18</font></i>: 
<font color="Maroon" title="c16">OO</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">PQ</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E35:28"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <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"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">P</font></span>)</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 NAME="E36:28"/><b>then </b><i><font color="Green" title="E39">A19</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">OO</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">GPQ</font>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E34:28"><i><font color="Green" title="E38">A18</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>set </b><font color="Maroon" title="c17">Ex</font> =  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c15">GPQ</font></span>)</span>;<br/>
<b>take </b><font color="Maroon" title="c18">EX</font> = <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p2">(<span class="default"><a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c15">GPQ</font></span>)</span></span>)</span>;<br/>

<a NAME="E37:28"/>
(  <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c15">GPQ</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c15">GPQ</font> is <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a> &amp;  <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c15">GPQ</font> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a> )
 
<b>by </b><i><a class="txt" href="jordan5a.html#E31:28"><i><font color="Green" title="E36">A16</font></i></a>, <a class="txt" href="jordan5a.html#E36:28"><i><font color="Green" title="E39">A19</font></i></a>, <a class="ref" href="seq_4.html#D3" title="SEQ_4:def.3">SEQ_4:def 3</a>, <a class="ref" href="weierstr.html#D3" title="WEIERSTR:def.3">WEIERSTR:def 3</a></i>;<br/>
<a NAME="E38:28"/><b>then </b>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c15">GPQ</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c15">GPQ</font>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T31" title="RCOMP_1:th.31">RCOMP_1:31</a></i>;<br/>
<a NAME="E39:28"/><b>then </b><i><font color="Green" title="E40">A20</font></i>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c15">GPQ</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">GPQ</font>
 
<b>by </b><i><a class="ref" href="weierstr.html#D3" title="WEIERSTR:def.3">WEIERSTR:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c19">EX</font> = <font color="Maroon" title="c18">EX</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">P</font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">g1</font> = <font color="Maroon" title="c7">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<a NAME="E42:28"/><i><font color="Green" title="E41">A21</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <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"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">P</font></span>)</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 NAME="E43:28"/><i><font color="Green" title="E42">A22</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <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> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">P</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan5a.html#E12:28"><i><font color="Green" title="E27">A7</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E44:28"/>
<font color="Maroon" title="c7">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E15:28"><i><font color="Green" title="E29">A9</font></i></a>, <a class="ref" href="tops_2.html#T63" title="TOPS_2:th.63">TOPS_2:63</a></i>;<br/>
<a NAME="E45:28"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g1</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
 
<b>by </b><i><a class="txt" href="jordan5a.html#E43:28"><i><font color="Green" title="E42">A22</font></i></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></i>;<br/>
<a NAME="E46:28"/><b>then </b><i><font color="Green" title="E43">A23</font></i>: 
( <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g1</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  &amp; <font color="Maroon" title="c7">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 
<b>by </b><i><a class="txt" href="jordan5a.html#E15:28"><i><font color="Green" title="E29">A9</font></i></a>, <a class="ref" href="tops_2.html#T63" title="TOPS_2:th.63">TOPS_2:63</a>, <a class="ref" href="tops_2.html#T64" title="TOPS_2:th.64">TOPS_2:64</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">pq</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E48:28"/><i><font color="Green" title="E44">A24</font></i>: 
( <font color="Maroon" title="c21">pq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> &amp; <font color="Maroon" title="c21">pq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">PQ</font> &amp;  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c15">GPQ</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">pq</font> )
 <b>by </b><i><a class="txt" href="jordan5a.html#E39:28"><i><font color="Green" title="E40">A20</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E49:28"/><i><font color="Green" title="E45">A25</font></i>: 
( <font color="Maroon" title="c7">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p1</font> &amp; <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p2</font> &amp; <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p2">(<span class="default"><a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c15">GPQ</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">EX</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c15">GPQ</font></span>)</span> &amp;  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c15">GPQ</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="jordan5a.html#E7:28"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="jordan5a.html#E39:28"><i><font color="Green" title="E40">A20</font></i></a>, <a class="txt" href="jordan5a.html#E3"><i><font color="Green" title="E3">Lm2</font></i></a></i>;<br/>
<a NAME="E50:28"/>
for <font color="Olive" title="b1">t</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">t</font> &amp; <font color="Olive" title="b1">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c15">GPQ</font></span>)</span> holds <br/> not <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c22">t</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:28_1"/><i><font color="Green" title="E46">A26</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">t</font> &amp; <font color="Maroon" title="c22">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c15">GPQ</font></span>)</span> )
 ;<br/>

<a NAME="E2:28_1"/><b>then </b><i><font color="Green" title="E47">A27</font></i>: 
<font color="Maroon" title="c22">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<b>by </b><i><a class="txt" href="jordan5a.html#E49:28"><i><font color="Green" title="E45">A25</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:28_1"/><b>then </b><i><font color="Green" title="E48">A28</font></i>: 
<font color="Maroon" title="c22">t</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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="txt" href="jordan5a.html#E1:28_1"><i><font color="Green" title="E46">A26</font></i></a>, <a class="txt" href="jordan5a.html#E3"><i><font color="Green" title="E3">Lm2</font></i></a></i>;<br/>
<a NAME="E4:28_1"/><i><font color="Green" title="E49">A29</font></i>: 
<font color="Maroon" title="c22">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f</font>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E14:28"><i><font color="Green" title="E28">A8</font></i></a>, <a class="txt" href="jordan5a.html#E1:28_1"><i><font color="Green" title="E46">A26</font></i></a>, <a class="txt" href="jordan5a.html#E2:28_1"><i><font color="Green" title="E47">A27</font></i></a>, <a class="txt" href="jordan5a.html#E3"><i><font color="Green" title="E3">Lm2</font></i></a></i>;<br/>
<b>assume </b><a NAME="E5:28_1"/><i><font color="Green" title="E50">A30</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Q</font>
 ;<br/>

<a NAME="E6:28_1"/>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">t</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 <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">P</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E3:28_1"><i><font color="Green" title="E48">A28</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<a NAME="E7:28_1"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E8:28_1"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">PQ</font>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E5:28_1"><i><font color="Green" title="E50">A30</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E9:28_1"/><b>then </b><i><font color="Green" title="E51">A31</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">t</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">GPQ</font>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E42:28"><i><font color="Green" title="E41">A21</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E10:28_1"/>
<font color="Maroon" title="c7">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f1</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
 
<b>by </b><i><a class="txt" href="jordan5a.html#E15:28"><i><font color="Green" title="E29">A9</font></i></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></i>;<br/>
<a NAME="E11:28_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">t</font>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E15:28"><i><font color="Green" title="E29">A9</font></i></a>, <a class="txt" href="jordan5a.html#E4:28_1"><i><font color="Green" title="E49">A29</font></i></a>, <a class="ref" href="funct_1.html#T56" title="FUNCT_1:th.56">FUNCT_1:56</a></i>;<br/>
<a NAME="E12:28_1"/><b>then </b>
<font color="Maroon" title="c22">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c15">GPQ</font>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E9:28_1"><i><font color="Green" title="E51">A31</font></i></a>, <a class="ref" href="weierstr.html#D3" title="WEIERSTR:def.3">WEIERSTR:def 3</a></i>;<br/>
<b>hence </b><a NAME="E13:28_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan5a.html#E32:28"><i><font color="Green" title="E37">A17</font></i></a>, <a class="txt" href="jordan5a.html#E1:28_1"><i><font color="Green" title="E46">A26</font></i></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E51:28"/>
( <font color="Maroon" title="c18">EX</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> &amp; <font color="Maroon" title="c18">EX</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Q</font> &amp;  ex <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</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> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span> ex <font color="Olive" title="b2">s2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">g</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p1</font> &amp; <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p2</font> &amp; <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">EX</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">s2</font> &amp; <font color="Olive" title="b2">s2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; ( for <font color="Olive" title="b3">t</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">t</font> &amp; <font color="Olive" title="b3">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">s2</font> holds <br/> not <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Q</font> ) ) )
 <b>by </b><i><a class="txt" href="jordan5a.html#E46:28"><i><font color="Green" title="E43">A23</font></i></a>, <a class="txt" href="jordan5a.html#E48:28"><i><font color="Green" title="E44">A24</font></i></a>, <a class="txt" href="jordan5a.html#E49:28"><i><font color="Green" title="E45">A25</font></i></a>, <a class="ref" href="funct_1.html#T56" title="FUNCT_1:th.56">FUNCT_1:56</a></i>;<br/>


</div>
