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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="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<font color="Maroon">c<sub>2</sub></font>;<br/>





<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>;<br/>
<b>assume </b><a NAME="E1:21"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R1">=</a>  <a href="funct_3.html#K10">pr2</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:21"/><i><font color="Green">E13</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<b>thus </b> <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a><br/>

<b>thus </b> <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> = 
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="borsuk_3.html#E1:21"><i><font color="Green">E12</font></i></a>, <a class="ref" href="funct_3.html#T62">FUNCT_3:62</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
;<br/>

<b>thus </b><a NAME="E5:21"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="borsuk_3.html#E1:21"><i><font color="Green">E12</font></i></a>, <a class="ref" href="borsuk_3.html#T7" target="_self">Th7</a></i>;<br/>

<b>thus </b><a NAME="E6:21"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="borsuk_3.html#E1:21"><i><font color="Green">E12</font></i></a>, <a class="txt" href="borsuk_3.html#E2:21"><i><font color="Green">E13</font></i></a>, <a class="ref" href="yellow12.html#T40">YELLOW12:40</a></i>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>3</sub></font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <b>by </b><i><a class="ref" href="borsuk_3.html#T2" target="_self">Th2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1"><a href="funct_3.html#K14">&lt;:</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="funct_3.html#K14">:&gt;</a></span> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="yellow12.html#T41">YELLOW12:41</a></i>;<br/>
<a NAME="E10:21"/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> 
 
<b>by </b><i><a class="txt" href="borsuk_3.html#E1:21"><i><font color="Green">E12</font></i></a>, <a class="ref" href="borsuk_3.html#T9" target="_self">Th9</a></i>;<br/>
<b>hence </b><a NAME="E11:21"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="pre_topc.html#V5">continuous</a>
 ;<br/>


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