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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#L1">TopStruct</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>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</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>assume </b><b>that </b><br/><a NAME="E1:49"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R1">=</a>  <a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:49"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R1">=</a>  <a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E3:49"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E4:49"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 ;<br/>

<a NAME="E5:49"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R1">=</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="yellow14.html#E2:49"><i><font color="Green">E15</font></i></a></i>;<br/>
<a NAME="E6:49"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R1">=</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="yellow14.html#E1:49"><i><font color="Green">E14</font></i></a></i>;<br/>
<i><font color="Green">E20</font></i>: the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> = 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="yellow14.html#E1:49"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow14.html#E2:49"><i><font color="Green">E15</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E8:49"/>
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:49_2_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E3:49_2_1"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="yellow14.html#E1:49_2_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><a NAME="E4:49_2_1"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="yellow14.html#E4:49"><i><font color="Green">E17</font></i></a>, <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/><a NAME="E5:49_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="yellow14.html#E5:49"><i><font color="Green">E18</font></i></a>, <a class="ref" href="borsuk_1.html#T4">BORSUK_1:4</a></i>;<br/><b>hence </b><a NAME="E6:49_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="yellow14.html#E4:49_2_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:49_2"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E4:49_2"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="yellow14.html#E2:49_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E5:49_2"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="yellow14.html#E3:49"><i><font color="Green">E16</font></i></a>, <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>
<a NAME="E6:49_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="yellow14.html#E6:49"><i><font color="Green">E19</font></i></a>, <a class="txt" href="yellow14.html#E7:49"><i><font color="Green">E20</font></i></a>, <a class="ref" href="borsuk_1.html#T4">BORSUK_1:4</a></i>;<br/>
<b>hence </b><a NAME="E7:49_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="yellow14.html#E5:49_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:49"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>2</sub></font> #)
 <b>by </b><i><a class="txt" href="yellow14.html#E7:49"><i><font color="Green">E20</font></i></a></i>;<br/>


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