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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="fin_topo.html#L1">FT_Space_Str</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="fin_topo.html#V2">filled</a>  <a href="fin_topo.html#L1">FT_Space_Str</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>assume </b><a NAME="E1:15"/>
<font color="Maroon">c<sub>4</sub></font> <a href="fintopo4.html#R2" target="_self">is_continuous</a> 0
 ;<br/>

<a NAME="E2:15"/><b>then </b><i><font color="Green">E8</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K10">U_FT</a> <font color="Olive">b<sub>1</sub></font>,0</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Olive">b<sub>2</sub></font>,0
 
<b>by </b><i><a class="ref" href="fintopo4.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E3:15"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K10">U_FT</a> <font color="Olive">b<sub>1</sub></font>,0</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><a NAME="E1:15_1"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<a NAME="E2:15_1"/><i><font color="Green">E10</font></i>: 
 <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">c<sub>6</sub></font>,0
 
<b>by </b><i><a class="ref" href="fintopo3.html#T47">FINTOPO3:47</a></i>;<br/>
<a NAME="E3:15_1"/>
 <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="fintopo3.html#K7">Finf</a> <span class="p1">(<span class="default"><a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="fintopo3.html#D10">FINTOPO3:def 10</a></i>;<br/>
<a NAME="E4:15_1"/><b>then </b><i><font color="Green">E11</font></i>: 
 <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">c<sub>6</sub></font>,0 <a href="tarski.html#R1">c=</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="fintopo4.html#E2:15_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="fintopo3.html#T36">FINTOPO3:36</a></i>;<br/>
<a NAME="E5:15_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">c<sub>5</sub></font>,0</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">c<sub>6</sub></font>,0
 
<b>by </b><i><a class="txt" href="fintopo4.html#E2:15"><i><font color="Green">E8</font></i></a>, <a class="txt" href="fintopo4.html#E1:15_1"><i><font color="Green">E9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:15_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">c<sub>5</sub></font>,0</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="fintopo4.html#E4:15_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:15"/>
<font color="Maroon">c<sub>4</sub></font> <a href="fintopo4.html#R2" target="_self">is_continuous</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="fintopo4.html#D2" target="_self">Def2</a></i>;<br/>


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