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<b>let </b><font color="Maroon">T1</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">S1</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">T2</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">S2</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">R1</font> be    <a href="yellow_9.html#M3">Refinement</a> of <font color="Maroon">T1</font>,<font color="Maroon">S1</font>;<br/><b>let </b><font color="Maroon">R2</font> be    <a href="yellow_9.html#M3">Refinement</a> of <font color="Maroon">T2</font>,<font color="Maroon">S2</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T1</font>,<font color="Maroon">T2</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S1</font>,<font color="Maroon">S2</font>;<br/><b>let </b><font color="Maroon">h</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">R1</font>,<font color="Maroon">R2</font>;<br/>











<b>assume </b><a NAME="E1:35"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">h</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">h</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">g</font> )
 ;<br/>

<b>assume </b><a NAME="E2:35"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 ;<br/>

<b>assume </b><a NAME="E3:35"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">K</font> = the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T2</font> <a href="xboole_0.html#K2">\/</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S2</font> as    <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">R2</font> <b>by </b><i><a class="ref" href="yellow_9.html#D6">YELLOW_9:def 6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R2</font>;<br/><b>assume </b><a NAME="E1:35_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T2</font> or <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S2</font> )
 </span><b>by </b><i><a class="ref" href="waybel19.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1"><b>suppose </b></a><a NAME="E1:35_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T2</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">A1</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T2</font> ;<br/><a NAME="E3:35_1_1_1"/>
<font color="Maroon">A1</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><a NAME="E4:35_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">A1</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/><b>hence </b><a NAME="E5:35_1_1_1"/>
<font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2"><b>suppose </b></a><a NAME="E1:35_1_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S2</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">A1</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S2</font> ;<br/><a NAME="E3:35_1_1_2"/>
<font color="Maroon">A1</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><a NAME="E4:35_1_1_2"/><b>then </b>
( <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">A1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">R1</font> is    <a href="yellow_9.html#M3">Refinement</a> of <font color="Maroon">S1</font>,<font color="Maroon">T1</font> )
 <b>by </b><i>, <a class="ref" href="tops_2.html#T55">TOPS_2:55</a>, <a class="ref" href="yellow_9.html#T55">YELLOW_9:55</a></i>;<br/><b>hence </b><a NAME="E5:35_1_1_2"/>
<font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:35"/>
<font color="Maroon">h</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="yellow_9.html#T36">YELLOW_9:36</a></i>;<br/>


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