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<b>let </b><font color="Maroon" title="c1">FT1</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">FT2</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#V3" title="ORDERS_2:attr.3">filled</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c3">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">FT1</font>,<font color="Maroon" title="c2">FT2</font>;<br/>







<b>assume </b><a NAME="E1:14"/><i><font color="Green" title="E7">A1</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="fintopo4.html#R2" title="FINTOPO4:pred.2">is_continuous</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<a NAME="E2:14"/>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT1</font><br/> for <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">FT2</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">FT1</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Olive" title="b1">x</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Olive" title="b2">y</font>,<font color="Maroon" title="c3">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT1</font>;<br/><b>let </b><font color="Maroon" title="c6">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">FT2</font>;<br/>



<b>assume </b><a NAME="E1:14_1"/><i><font color="Green" title="E8">A2</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">FT1</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> )
 ;<br/>

<a NAME="E2:14_1"/><i><font color="Green" title="E9">A3</font></i>: 
 <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c6">y</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="fintopo3.html#T47" title="FINTOPO3:th.47">FINTOPO3:47</a></i>;<br/>
<a NAME="E3:14_1"/>
 <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fintopo3.html#K7" title="FINTOPO3:func.7">Finf</a> <span class="p1">(<span class="default"><a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c6">y</font></span>)</span>,<font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="ref" href="fintopo3.html#D10" title="FINTOPO3:def.10">FINTOPO3:def 10</a></i>;<br/>
<a NAME="E4:14_1"/><b>then </b><i><font color="Green" title="E10">A4</font></i>: 
 <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c6">y</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="txt" href="fintopo4.html#E2:14_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="fintopo3.html#T36" title="FINTOPO3:th.36">FINTOPO3:36</a></i>;<br/>
<a NAME="E5:14_1"/>
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c5">x</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c6">y</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="fintopo4.html#E1:14"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="fintopo4.html#E1:14_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="fintopo4.html#D2" target="_self" title="FINTOPO4:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E6:14_1"/>
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c5">x</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="fintopo4.html#E4:14_1"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:14"/>
<font color="Maroon" title="c4">f</font> <a href="fintopo4.html#R2" title="FINTOPO4:pred.2">is_continuous</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="ref" href="fintopo4.html#D2" target="_self" title="FINTOPO4:def.2">Def2</a></i>;<br/>


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