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<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>;<br/><b>let </b><font color="Maroon" title="c4">ft</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</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>,<span class="p1">(<span class="default"><a href="normsp_2.html#K3" title="NORMSP_2:func.3">TopSpaceNorm</a> <font color="Maroon" title="c2">Y</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c5">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="c6">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="E1:27"/><i><font color="Green" title="E22">A1</font></i>: 
( <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">ft</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">xt</font> )
 ;<br/>

<a NAME="E2:27"/><b>then </b><i><font color="Green" title="E23">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">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 <font color="Maroon" title="c1">X</font>
 
<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="E3:27"/><b>then </b><i><font color="Green" title="E24">A3</font></i>: 
<font color="Maroon" title="c4">ft</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">xt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x</font>
 
<b>by </b><i><a class="txt" href="normsp_2.html#E1:27"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:27_1"/><i><font color="Green" title="E25">A4</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c5">x</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">G</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c4">ft</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">xt</font>;<br/><b>reconsider </b><font color="Maroon" title="c8">N1</font> = <font color="Maroon" title="c7">G</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> ;<br/><a NAME="E2:27_1_1"/>
<font color="Maroon" title="c8">N1</font> is    <a href="nfcont_1.html#M1" title="NFCONT_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="normsp_2.html#E3:27"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="normsp_2.html#T17" target="_self" title="NORMSP_2:th.17">Th17</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">N</font> being    <a href="nfcont_1.html#M1" title="NFCONT_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c5">x</font><b> such that </b><br/><a NAME="E4:27_1_1"/><i><font color="Green" title="E26">A5</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">N1</font>
 <b>by </b><i><a class="txt" href="normsp_2.html#E1:27_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="nfcont_1.html#T17" title="NFCONT_1:th.17">NFCONT_1:17</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">H</font> = <font color="Maroon" title="c9">N</font> as    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c6">xt</font> <b>by </b><i><a class="txt" href="normsp_2.html#E1:27"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="normsp_2.html#T17" target="_self" title="NORMSP_2:th.17">Th17</a></i>;<br/><b>take </b><font color="Maroon" title="c11">H</font> = <font color="Maroon" title="c10">H</font>;<br/><b>thus </b><a NAME="E6:27_1_1"/>
<font color="Maroon" title="c4">ft</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c11">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">G</font>
 <b>by </b><i><a class="txt" href="normsp_2.html#E4:27_1_1"><i><font color="Green" title="E26">A5</font></i></a>, <a class="txt" href="normsp_2.html#E1:27"><i><font color="Green" title="E22">A1</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:27_1"/>
<font color="Maroon" title="c4">ft</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c6">xt</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#D2" title="TMAP_1:def.2">TMAP_1:def 2</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E5:27"/><i><font color="Green" title="E25">A6</font></i>: 
<font color="Maroon" title="c4">ft</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c6">xt</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">N1</font> be    <a href="nfcont_1.html#M1" title="NFCONT_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x</font>;<br/><b>reconsider </b><font color="Maroon" title="c8">G</font> = <font color="Maroon" title="c7">N1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> ;<br/><a NAME="E2:27_2"/>
<font color="Maroon" title="c8">G</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c4">ft</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">xt</font>
 <b>by </b><i><a class="txt" href="normsp_2.html#E3:27"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="normsp_2.html#T17" target="_self" title="NORMSP_2:th.17">Th17</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">H</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c6">xt</font><b> such that </b><br/><a NAME="E4:27_2"/><i><font color="Green" title="E26">A7</font></i>: 
<font color="Maroon" title="c4">ft</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">G</font>
 <b>by </b><i><a class="txt" href="normsp_2.html#E5:27"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="tmap_1.html#D2" title="TMAP_1:def.2">TMAP_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">N</font> = <font color="Maroon" title="c9">H</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/><b>reconsider </b><font color="Maroon" title="c11">N</font> = <font color="Maroon" title="c10">N</font> as    <a href="nfcont_1.html#M1" title="NFCONT_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c5">x</font> <b>by </b><i><a class="txt" href="normsp_2.html#E1:27"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="normsp_2.html#T17" target="_self" title="NORMSP_2:th.17">Th17</a></i>;<br/><b>take </b><font color="Maroon" title="c12">N</font> = <font color="Maroon" title="c11">N</font>;<br/><b>thus </b><a NAME="E7:27_2"/>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c12">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">N1</font>
 <b>by </b><i><a class="txt" href="normsp_2.html#E4:27_2"><i><font color="Green" title="E26">A7</font></i></a>, <a class="txt" href="normsp_2.html#E1:27"><i><font color="Green" title="E22">A1</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:27"/>
<font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="normsp_2.html#E2:27"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="nfcont_1.html#T17" title="NFCONT_1:th.17">NFCONT_1:17</a></i>;<br/>


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