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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:21"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel18.html#V2" target="_self">injective</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E2:21"/>
<font color="Maroon">c<sub>2</sub></font> <a href="borsuk_1.html#R1">is_a_retract_of</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being  <a href="pre_topc.html#V5">continuous</a> <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><b> such that </b><br/><a NAME="E4:21"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="borsuk_1.html#V3">being_a_retraction</a>
 <b>by </b><i><a class="ref" href="borsuk_1.html#D20">BORSUK_1:def 20</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="tmap_1.html#K4">incl</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E5:21"/><i><font color="Green">E14</font></i>: 
 <a href="tmap_1.html#K4">incl</a> <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T98">TMAP_1:98</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>5</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel18.html#D5" target="_self">WAYBEL18:def 5</a><br/><b>let </b><font color="Maroon">c<sub>6</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>2</sub></font>;<br/>



<b>assume </b><a NAME="E6:21"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E7:21"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><a href="tmap_1.html#K4">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E9:21"/>
<font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="waybel18.html#E5:21"><i><font color="Green">E14</font></i></a>, <a class="txt" href="waybel18.html#E6:21"><i><font color="Green">E15</font></i></a>, <a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E11:21"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E12:21"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="waybel18.html#E1:21"><i><font color="Green">E12</font></i></a>, <a class="txt" href="waybel18.html#E7:21"><i><font color="Green">E16</font></i></a>, <a class="ref" href="waybel18.html#D5" target="_self">Def5</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font>;<br/>

<b>thus </b><a NAME="E13:21"/>
<font color="Maroon">c<sub>10</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="waybel18.html#E11:21"><i><font color="Green">E17</font></i></a>, <a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>

<a NAME="E14:21"/><i><font color="Green">E19</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E7:21"><i><font color="Green">E16</font></i></a>, <a class="ref" href="borsuk_1.html#T35">BORSUK_1:35</a></i>;<br/>
<a NAME="E15:21"/><i><font color="Green">E20</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</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="ref" href="borsuk_1.html#T35">BORSUK_1:35</a></i>;<br/>
<i><font color="Green">E21</font></i>: <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> = 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#K3">/\</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/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="waybel18.html#E14:21"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E17:21"/><i><font color="Green">E22</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</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/>
<a NAME="E18:21"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:21_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E2:21_2"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>
 
;<br/>
<a NAME="E3:21_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E1:21_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:21_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel18.html#E15:21"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E6:21_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E4:21"><i><font color="Green">E13</font></i></a>, <a class="txt" href="waybel18.html#E3:21_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="borsuk_1.html#D19">BORSUK_1:def 19</a></i>;<br/>
<i><font color="Green">E27</font></i>: <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> = 
<span class="p1">(<span class="default"><a href="tmap_1.html#K4">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="waybel18.html#E2:21_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="waybel18.html#E3:21_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="tmap_1.html#T95">TMAP_1:95</a></i>
;<br/>
<a NAME="E8:21_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E12:21"><i><font color="Green">E18</font></i></a>, <a class="txt" href="waybel18.html#E2:21_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><a NAME="E9:21_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="waybel18.html#E14:21"><i><font color="Green">E19</font></i></a>, <a class="txt" href="waybel18.html#E2:21_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="waybel18.html#E6:21_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="waybel18.html#E7:21_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E19:21"/>
<font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="waybel18.html#E16:21"><i><font color="Green">E21</font></i></a>, <a class="txt" href="waybel18.html#E17:21"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>


</div>
