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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="tdlat_3.html#V3" title="TDLAT_3:attr.3">almost_discrete</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">X0</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="tdlat_3.html#V1" title="TDLAT_3:attr.1">discrete</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>;<br/>



<b>consider </b><font color="Maroon" title="c3">Z0</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E2:127"/><i><font color="Green" title="E60">A1</font></i>: 
<font color="Maroon" title="c2">X0</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">Z0</font>
 <b>and </b><br/><a NAME="E3:127"/><i><font color="Green" title="E61">A2</font></i>: 
<font color="Maroon" title="c3">Z0</font> is <a href="tex_2.html#V5" title="TEX_2:attr.5">maximal_discrete</a>
 <b>by </b><i><a class="ref" href="tex_2.html#T67" target="_self" title="TEX_2:th.67">Th67</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">Z0</font> = <font color="Maroon" title="c3">Z0</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a> <a href="tex_2.html#V5" title="TEX_2:attr.5">maximal_discrete</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="tex_2.html#E3:127"><i><font color="Green" title="E61">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">Z1</font> = <font color="Maroon" title="c4">Z0</font> as  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/>
<b>reconsider </b><font color="Maroon" title="c6">Z1</font> = <font color="Maroon" title="c5">Z1</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="tdlat_3.html#V1" title="TDLAT_3:attr.1">discrete</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">X1</font> = <font color="Maroon" title="c2">X0</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="c6">Z1</font> <b>by </b><i><a class="txt" href="tex_2.html#E2:127"><i><font color="Green" title="E60">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">g</font> being  <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="c6">Z1</font>,<font color="Maroon" title="c7">X1</font><b> such that </b><br/><a NAME="E9:127"/><i><font color="Green" title="E62">A3</font></i>: 
<font color="Maroon" title="c8">g</font> <a href="borsuk_1.html#NR1" title="BORSUK_1:NR.1">is_a_retraction</a> 
 <b>by </b><i><a class="ref" href="tex_2.html#T72" target="_self" title="TEX_2:th.72">Th72</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">g</font> = <font color="Maroon" title="c8">g</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="c4">Z0</font>,<font color="Maroon" title="c2">X0</font> ;<br/>
<b>consider </b><font color="Maroon" title="c10">h</font> being  <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="c1">X</font>,<font color="Maroon" title="c4">Z0</font><b> such that </b><br/><a NAME="E12:127"/><i><font color="Green" title="E63">A4</font></i>: 
<font color="Maroon" title="c10">h</font> <a href="borsuk_1.html#NR1" title="BORSUK_1:NR.1">is_a_retraction</a> 
 <b>by </b><i><a class="ref" href="tex_2.html#T74" target="_self" title="TEX_2:th.74">Th74</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">r</font> = <font color="Maroon" title="c9">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">h</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="c1">X</font>,<font color="Maroon" title="c2">X0</font> <b>by </b><i><a class="ref" href="tops_2.html#T58" title="TOPS_2:th.58">TOPS_2:58</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c11">r</font>
;<br/>

<a NAME="E14:127"/>
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">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="c2">X0</font> holds <br/><font color="Maroon" title="c11">r</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>assume </b><a NAME="E1:127_1"/><i><font color="Green" title="E64">A5</font></i>: 
<font color="Maroon" title="c12">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="c2">X0</font>
 ;<br/>

<a NAME="E2:127_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</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="c6">Z1</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T35" title="BORSUK_1:th.35">BORSUK_1:35</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">y</font> = <font color="Maroon" title="c12">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c6">Z1</font> <b>by </b><i><a class="txt" href="tex_2.html#E1:127_1"><i><font color="Green" title="E64">A5</font></i></a></i>;<br/>
<a NAME="E4:127_1"/>
<font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y</font>
 
<b>by </b><i><a class="txt" href="tex_2.html#E9:127"><i><font color="Green" title="E62">A3</font></i></a>, <a class="txt" href="tex_2.html#E1:127_1"><i><font color="Green" title="E64">A5</font></i></a>, <a class="ref" href="borsuk_1.html#D19" title="BORSUK_1:def.19">BORSUK_1:def 19</a></i>;<br/>
<a NAME="E5:127_1"/><b>then </b>
<font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font>
 
<b>by </b><i><a class="txt" href="tex_2.html#E12:127"><i><font color="Green" title="E63">A4</font></i></a>, <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="E6:127_1"/>
<font color="Maroon" title="c11">r</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E15:127"/>
<font color="Maroon" title="c11">r</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/>


</div>
