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<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">U</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">Ut</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" title="NORMSP_2:func.3">TopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c4">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c5">xt</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" title="NORMSP_2:func.3">TopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span>;<br/>









<b>assume </b><a NAME="E1:26"/><i><font color="Green" title="E21">A1</font></i>: 
( <font color="Maroon" title="c2">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Ut</font> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">xt</font> )
 ;<br/>

<div><i><font color="Green" title="E22">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_1"/>
<font color="Maroon" title="c3">Ut</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c5">xt</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">Vt</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" title="NORMSP_2:func.3">TopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span><b> such that </b><br/><a NAME="E3:26_1"/><i><font color="Green" title="E23">A3</font></i>: 
( <font color="Maroon" title="c6">Vt</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c6">Vt</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Ut</font> &amp; <font color="Maroon" title="c5">xt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Vt</font> )
 <b>by </b><i><a class="ref" href="connsp_2.html#T8" title="CONNSP_2:th.8">CONNSP_2:8</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:26_1"/><i><font color="Green" title="E24">A4</font></i>: 
( <font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> : <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Olive" title="b1">y</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">r</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">Vt</font> )
 <b>by </b><i><a class="txt" href="normsp_2.html#E3:26_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="normsp_2.html#T7" target="_self" title="NORMSP_2:th.7">Th7</a>, <a class="txt" href="normsp_2.html#E1:26"><i><font color="Green" title="E21">A1</font></i></a></i>;<br/><a NAME="E6:26_1"/><i><font color="Green" title="E25">A5</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> : <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Olive" title="b1">y</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">r</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">U</font>
 <b>by </b><i><a class="txt" href="normsp_2.html#E1:26"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="normsp_2.html#E5:26_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="txt" href="normsp_2.html#E3:26_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E7:26_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> : <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Olive" title="b1">y</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">r</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> : <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">y</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c4">x</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">r</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="normsp_2.html#E24"><i><font color="Green" title="E19">Lm5</font></i></a></i>;<br/><b>hence </b><a NAME="E8:26_1"/>
<font color="Maroon" title="c2">U</font> is    <a href="nfcont_1.html#M1" title="NFCONT_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="normsp_2.html#E5:26_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="nfcont_1.html#D3" title="NFCONT_1:def.3">NFCONT_1:def 3</a>, <a class="txt" href="normsp_2.html#E6:26_1"><i><font color="Green" title="E25">A5</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_2"/>
<font color="Maroon" title="c2">U</font> is    <a href="nfcont_1.html#M1" title="NFCONT_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c4">x</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:26_2"/><i><font color="Green" title="E23">A6</font></i>: 
( <font color="Maroon" title="c6">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> : <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">y</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c4">x</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">r</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">U</font> )
 <b>by </b><i><a class="ref" href="nfcont_1.html#D3" title="NFCONT_1:def.3">NFCONT_1:def 3</a></i>;<br/><a NAME="E4:26_2"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> : <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">y</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c4">x</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">r</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="c1">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:26_2_1_1"/>
<font color="Maroon" title="c7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> : <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">y</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c4">x</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">r</font> </span>}</span> 
 ;<br/><a NAME="E2:26_2_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">z</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">z</font> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">z</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c4">x</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">r</font> )
 ;<br/><b>hence </b><a NAME="E3:26_2_1_1"/>
<font color="Maroon" title="c7">s</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="c1">X</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:26_2_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> : <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">y</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c4">x</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">r</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="c1">X</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c7">Vt</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> : <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">y</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c4">x</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">r</font> </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" title="NORMSP_2:func.3">TopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span> ;<br/><a NAME="E6:26_2"/>
<font color="Maroon" title="c7">Vt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> : <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Olive" title="b1">y</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">r</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="normsp_2.html#E24"><i><font color="Green" title="E19">Lm5</font></i></a></i>;<br/><a NAME="E7:26_2"/><b>then </b><i><font color="Green" title="E24">A7</font></i>: 
<font color="Maroon" title="c7">Vt</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="normsp_2.html#T8" target="_self" title="NORMSP_2:th.8">Th8</a></i>;<br/><a NAME="E8:26_2"/>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c4">x</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="normsp_1.html#T10" title="NORMSP_1:th.10">NORMSP_1:10</a></i>;<br/><a NAME="E9:26_2"/><b>then </b>
<font color="Maroon" title="c5">xt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Vt</font>
 <b>by </b><i><a class="txt" href="normsp_2.html#E1:26"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="normsp_2.html#E3:26_2"><i><font color="Green" title="E23">A6</font></i></a></i>;<br/><b>hence </b><a NAME="E10:26_2"/>
<font color="Maroon" title="c3">Ut</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c5">xt</font>
 <b>by </b><i><a class="txt" href="normsp_2.html#E7:26_2"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="normsp_2.html#E3:26_2"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="normsp_2.html#E1:26"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="connsp_2.html#T8" title="CONNSP_2:th.8">CONNSP_2:8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:26"/>
( <font color="Maroon" title="c2">U</font> is    <a href="nfcont_1.html#M1" title="NFCONT_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c4">x</font> iff <font color="Maroon" title="c3">Ut</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c5">xt</font> )
 <b>by </b><i><a class="txt" href="normsp_2.html#E2:26"><i><font color="Green" title="E22">A2</font></i></a></i>;<br/>


</div>
