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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="tsep_1.html#V1">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>











<b>assume </b><a NAME="E1:68"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>hence </b><a NAME="E2:68"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>5</sub></font> implies <font color="Maroon">c<sub>3</sub></font> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="tmap_1.html#T64" target="_self">Th64</a></i>;<br/>

<b>thus </b><a NAME="E3:68"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>6</sub></font> implies <font color="Maroon">c<sub>3</sub></font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>5</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<a NAME="E2:68_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T16">TSEP_1:16</a></i>;<br/>
<b>assume </b><a NAME="E3:68_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>hence </b><a NAME="E4:68_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:68"><i><font color="Green">E52</font></i></a>, <a class="txt" href="tmap_1.html#E2:68_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="tmap_1.html#T66" target="_self">Th66</a></i>;<br/>


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


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