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<b>let </b><font color="Maroon" title="c1">R</font>, <font color="Maroon" title="c2">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:78_1"/>
<font color="Maroon" title="c1">R</font> <a href="waybel18.html#R1" title="WAYBEL18:pred.1">is_Retract_of</a> <font color="Maroon" title="c2">T</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c3">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c2">T</font><b> such that </b><br/><a NAME="E3:78_1"/><i><font color="Green" title="E47">A1</font></i>: 
( <font color="Maroon" title="c3">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">f</font> &amp;  <a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">R</font> <a href="t_0topsp.html#R1" title="T_0TOPSP:pred.1">are_homeomorphic</a>  )
 <b>by </b><i><a class="ref" href="waybel18.html#D8" title="WAYBEL18:def.8">WAYBEL18:def 8</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c4">S</font> =  <a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c3">f</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c2">T</font> ;<br/><b>take </b><font color="Maroon" title="c5">S</font> = <font color="Maroon" title="c4">S</font>;<br/><a NAME="E5:78_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="ref" href="waybel18.html#D7" title="WAYBEL18:def.7">WAYBEL18:def 7</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c6">f</font> = <font color="Maroon" title="c3">f</font> as  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c5">S</font> <b>by </b><i><a class="txt" href="yellow16.html#E3:78_1"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="waybel18.html#T11" title="WAYBEL18:th.11">WAYBEL18:11</a></i>;<br/><a NAME="E7:78_1"/>
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">S</font> &amp;  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font> )
 ;<br/><a NAME="E8:78_1"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D9" title="PRE_TOPC:def.9">PRE_TOPC:def 9</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c7">rf</font> =  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">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" title="c8">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T</font>;<br/><b>assume </b><a NAME="E1:78_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">S</font>
 ;<br/><a NAME="E2:78_1_1"/><b>then </b>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c7">rf</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel18.html#D6" title="WAYBEL18:def.6">WAYBEL18:def 6</a></i>;<br/><a NAME="E3:78_1_1"/><b>then </b>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/><a NAME="E4:78_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E5:78_1_1"/>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="txt" href="yellow16.html#E3:78_1"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/></div><b>end;</b></div><a NAME="E11:78_1"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="borsuk_1.html#NR1" title="BORSUK_1:NR.1">is_a_retraction</a> 
 <b>by </b><i><a class="ref" href="borsuk_1.html#D19" title="BORSUK_1:def.19">BORSUK_1:def 19</a></i>;<br/><b>hence </b><a NAME="E12:78_1"/>
( <font color="Maroon" title="c5">S</font> <a href="borsuk_1.html#R1" title="BORSUK_1:pred.1">is_a_retract_of</a> <font color="Maroon" title="c2">T</font> &amp; <font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">R</font> <a href="t_0topsp.html#R1" title="T_0TOPSP:pred.1">are_homeomorphic</a>  )
 <b>by </b><i><a class="txt" href="yellow16.html#E3:78_1"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="borsuk_1.html#D20" title="BORSUK_1:def.20">BORSUK_1:def 20</a></i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon" title="c3">S</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c2">T</font><b> such that </b><a NAME="E3:78"/><i><font color="Green" title="E47">A2</font></i>: 
( <font color="Maroon" title="c3">S</font> <a href="borsuk_1.html#R1" title="BORSUK_1:pred.1">is_a_retract_of</a> <font color="Maroon" title="c2">T</font> &amp; <font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c1">R</font> <a href="t_0topsp.html#R1" title="T_0TOPSP:pred.1">are_homeomorphic</a>  )
 ;<br/>

<b>thus </b><a NAME="E4:78"/>
<font color="Maroon" title="c1">R</font> <a href="waybel18.html#R1" title="WAYBEL18:pred.1">is_Retract_of</a> <font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="yellow16.html#E3:78"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="yellow16.html#T55" target="_self" title="YELLOW16:th.55">Th55</a>, <a class="ref" href="yellow16.html#T58" target="_self" title="YELLOW16:th.58">Th58</a></i>;<br/>


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