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<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</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>7</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>;<br/>



<b>assume </b><a NAME="E1:22"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>5</sub></font>;<br/>

<b>given </b><font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span><b> such that </b><a NAME="E2:22"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">c<sub>5</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>9</sub></font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_1.html#D6">CONNSP_1:def 6</a><br/>

<a NAME="E3:22"/>
(  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E4:22"/><b>then </b><i><font color="Green">E19</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="jordan24.html#E2:22"><i><font color="Green">E18</font></i></a>, <a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<a NAME="E6:22"/><i><font color="Green">E20</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1"/>
( <font color="Maroon">c<sub>9</sub></font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">c<sub>9</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1"><b>suppose </b></a><a NAME="E1:22_1_1"/>
<font color="Maroon">c<sub>9</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:22_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#T149">RELAT_1:149</a></i>;<br/><a NAME="E3:22_1_1"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="jordan24.html#E4:22"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><b>hence </b><a NAME="E4:22_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>8</sub></font> <a href="connsp_1.html#R4">is_a_component_of</a> <font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="jordan1k.html#T6">JORDAN1K:6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_2"><b>suppose </b></a><a NAME="E1:22_1_2"/><i><font color="Green">E21</font></i>: 
not <font color="Maroon">c<sub>9</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:22_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="jordan24.html#E6:22"><i><font color="Green">E20</font></i></a>, <a class="ref" href="relat_1.html#T152">RELAT_1:152</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> as  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a> <b>by </b><i><a class="txt" href="jordan24.html#E4:22"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan24.html#E1:22_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>11</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_1.html#D6">CONNSP_1:def 6</a><br/><b>thus </b><a NAME="E4:22_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> <b>by </b><i><a class="ref" href="jordan24.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E6:22_1_2"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> is <a href="tops_2.html#V3">being_homeomorphism</a> )
 <b>by </b><i><a class="txt" href="jordan24.html#E2:22"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan24.html#E4:22"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan24.html#E1:22"><i><font color="Green">E17</font></i></a>, <a class="ref" href="jordan24.html#T14" target="_self">Th14</a>, <a class="ref" href="extens_1.html#T1">EXTENS_1:1</a></i>;<br/><b>hence </b><a NAME="E7:22_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">c<sub>6</sub></font> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan24.html#E2:22"><i><font color="Green">E18</font></i></a>, <a class="ref" href="jordan24.html#T11" target="_self">Th11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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