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<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">X1</font>, <font color="Maroon" title="c4">X0</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c5">g</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">X0</font>,<font color="Maroon" title="c2">Y</font>;<br/>









<b>assume </b><a NAME="E1:82"/><i><font color="Green" title="E68">A1</font></i>: 
<font color="Maroon" title="c3">X1</font> is  <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">X0</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c6">x0</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">X0</font>;<br/><b>let </b><font color="Maroon" title="c7">x1</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">X1</font>;<br/>



<b>assume </b><a NAME="E2:82"/><i><font color="Green" title="E69">A2</font></i>: 
<font color="Maroon" title="c6">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x1</font>
 ;<br/>

<b>hence </b><a NAME="E3:82"/>
( <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c6">x0</font> implies <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x1</font> )
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:82"><i><font color="Green" title="E68">A1</font></i></a>, <a class="ref" href="tmap_1.html#T81" target="_self" title="TMAP_1:th.81">Th81</a></i>;<br/>

<b>thus </b><a NAME="E4:82"/>
( <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x1</font> implies <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c6">x0</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c8">A</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">X0</font> <b>by </b><i><a class="txt" href="tmap_1.html#E1:82"><i><font color="Green" title="E68">A1</font></i></a>, <a class="ref" href="tsep_1.html#T1" title="TSEP_1:th.1">TSEP_1:1</a></i>;<br/>
<a NAME="E2:82_1"/><i><font color="Green" title="E70">A3</font></i>: 
<font color="Maroon" title="c8">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="tmap_1.html#E1:82"><i><font color="Green" title="E68">A1</font></i></a>, <a class="ref" href="tsep_1.html#T16" title="TSEP_1:th.16">TSEP_1:16</a></i>;<br/>
<b>assume </b><a NAME="E3:82_1"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x1</font>
 ;<br/>

<b>hence </b><a NAME="E4:82_1"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c6">x0</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:82"><i><font color="Green" title="E68">A1</font></i></a>, <a class="txt" href="tmap_1.html#E2:82"><i><font color="Green" title="E69">A2</font></i></a>, <a class="txt" href="tmap_1.html#E2:82_1"><i><font color="Green" title="E70">A3</font></i></a>, <a class="ref" href="tmap_1.html#T84" target="_self" title="TMAP_1:th.84">Th84</a></i>;<br/>


</div><b>end;</b></div>


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