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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/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:78_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="waybel18.html#R1">is_Retract_of</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:78_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="waybel18.html#K7">Image</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="t_0topsp.html#R1">are_homeomorphic</a>  )
 <b>by </b><i><a class="ref" href="waybel18.html#D8">WAYBEL18:def 8</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="waybel18.html#K7">Image</a> <font color="Maroon">c<sub>3</sub></font> as  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/><b>take </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font>;<br/><a NAME="E5:78_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R1">=</a>  <a href="waybel18.html#K8">corestr</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="waybel18.html#D7">WAYBEL18:def 7</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <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>,<font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="yellow16.html#E3:78_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="waybel18.html#T11">WAYBEL18:11</a></i>;<br/><a NAME="E7:78_1"/>
(  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>5</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> &amp;  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <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/><a NAME="E8:78_1"/><b>then </b>
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>2</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><a NAME="E1:78_1_1"/>
<font color="Maroon">c<sub>8</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="E2:78_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel18.html#D6">WAYBEL18:def 6</a></i>;<br/><a NAME="E3:78_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/><a NAME="E4:78_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</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="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/><a NAME="E5:78_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <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> &amp; <font color="Maroon">c<sub>8</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="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E6:78_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="yellow16.html#E3:78_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/></div><b>end;</b></div><a NAME="E11:78_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="borsuk_1.html#NR1" target="_self">is_a_retraction</a> 
 <b>by </b><i><a class="ref" href="borsuk_1.html#D19">BORSUK_1:def 19</a></i>;<br/><b>hence </b><a NAME="E12:78_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="borsuk_1.html#R1">is_a_retract_of</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="t_0topsp.html#R1">are_homeomorphic</a>  )
 <b>by </b><i><a class="txt" href="yellow16.html#E3:78_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="borsuk_1.html#D20">BORSUK_1:def 20</a></i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">c<sub>3</sub></font> being  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><a NAME="E2:78"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="borsuk_1.html#R1">is_a_retract_of</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="t_0topsp.html#R1">are_homeomorphic</a>  )
 ;<br/>

<a NAME="E3:78"/>
<font color="Maroon">c<sub>3</sub></font> <a href="waybel18.html#R1">is_Retract_of</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="yellow16.html#E2:78"><i><font color="Green">E48</font></i></a>, <a class="ref" href="yellow16.html#T58" target="_self">Th58</a></i>;<br/>
<b>hence </b><a NAME="E4:78"/>
<font color="Maroon">c<sub>1</sub></font> <a href="waybel18.html#R1">is_Retract_of</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="yellow16.html#E2:78"><i><font color="Green">E48</font></i></a>, <a class="ref" href="yellow16.html#T55" target="_self">Th55</a></i>;<br/>


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