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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#V2" title="STRUCT_0:attr.2">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#V2" title="STRUCT_0:attr.2">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="E1:84"/><i><font color="Green" title="E62">A1</font></i>: 
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X1</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">X1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="E2:84"/><i><font color="Green" title="E63">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>reconsider </b><font color="Maroon" title="c8">Y1</font> =  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X1</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">X1</font> #) as   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a> <b>by </b><i><a class="txt" href="tmap_1.html#E12"><i><font color="Green" title="E4">Lm1</font></i></a></i>;<br/>
<a NAME="E4:84"/><i><font color="Green" title="E64">A3</font></i>: 
<font color="Maroon" title="c8">Y1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">X0</font>
 
<b>by </b><i><a class="txt" href="tmap_1.html#E1:84"><i><font color="Green" title="E62">A1</font></i></a>, <a class="ref" href="tsep_1.html#T2" title="TSEP_1:th.2">TSEP_1:2</a></i>;<br/>
<a NAME="E5:84"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X1</font> <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="c4">X0</font>
 
<b>by </b><i><a class="txt" href="tmap_1.html#E1:84"><i><font color="Green" title="E62">A1</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">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> ;<br/>
<a NAME="E7:84"/>
<font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c4">X0</font>
 
<b>by </b><i><a class="txt" href="tmap_1.html#E1:84"><i><font color="Green" title="E62">A1</font></i></a></i>;<br/>
<a NAME="E8:84"/><b>then </b>
<font color="Maroon" title="c9">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T53" title="TOPS_1:th.53">TOPS_1:53</a></i>;<br/>
<a NAME="E9:84"/><b>then </b><i><font color="Green" title="E65">A4</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>
 
<b>by </b><i><a class="txt" href="tmap_1.html#E4:84"><i><font color="Green" title="E64">A3</font></i></a>, <a class="ref" href="tmap_1.html#T12" target="_self" title="TMAP_1:th.12">Th12</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="E10:84"/>
<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="E11:84"/>
<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#E2:84"><i><font color="Green" title="E63">A2</font></i></a>, <a class="txt" href="tmap_1.html#E9:84"><i><font color="Green" title="E65">A4</font></i></a>, <a class="ref" href="tmap_1.html#T86" target="_self" title="TMAP_1:th.86">Th86</a></i>;<br/>


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