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<b>let </b><font color="Maroon" title="c5">S</font>, <font color="Maroon" title="c6">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/>

<b>given </b><font color="Maroon" title="c7">h</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c6">T</font><b> such that </b><a NAME="E2:11"/><i><font color="Green" title="E6">A1</font></i>: 
<font color="Maroon" title="c7">h</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="t_0topsp.html#D1" title="T_0TOPSP:def.1">T_0TOPSP:def 1</a><br/>

<b>assume </b><a NAME="E3:11"/><i><font color="Green" title="E7">A2</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c5">S</font> holds  <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_2.html#D3" title="BORSUK_2:def.3">BORSUK_2:def 3</a><br/>

<b>let </b><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">b</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c6">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_2.html#D3" title="BORSUK_2:def.3">BORSUK_2:def 3</a><br/>

<a NAME="E4:11"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 
<b>by </b><i><a class="txt" href="topalg_3.html#E3:11"><i><font color="Green" title="E7">A2</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a> ,<font color="Maroon" title="c5">S</font><b> such that </b><br/><a NAME="E6:11"/><i><font color="Green" title="E8">A3</font></i>: 
<font color="Maroon" title="c10">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E7:11"/><i><font color="Green" title="E9">A4</font></i>: 
<font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">a</font>
 <b>and </b><br/><a NAME="E8:11"/><i><font color="Green" title="E10">A5</font></i>: 
<font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font>
 <b>by </b><i><a class="ref" href="borsuk_2.html#D1" title="BORSUK_2:def.1">BORSUK_2:def 1</a></i>;<br/>
<b>take </b><font color="Maroon" title="c11">g</font> = <font color="Maroon" title="c7">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">f</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_2.html#D1" title="BORSUK_2:def.1">BORSUK_2:def 1</a><br/>

<a NAME="E9:11"/>
<font color="Maroon" title="c7">h</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="topalg_3.html#E2:11"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E10:11"/>
<font color="Maroon" title="c11">g</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="topalg_3.html#E6:11"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="tops_2.html#T58" title="TOPS_2:th.58">TOPS_2:58</a></i>;<br/>

<a NAME="E11:11"/><i><font color="Green" title="E11">A6</font></i>: 
<font color="Maroon" title="c7">h</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="topalg_3.html#E2:11"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E12:11"/><i><font color="Green" title="E12">A7</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c6">T</font>
 
<b>by </b><i><a class="txt" href="topalg_3.html#E2:11"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c11">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<font color="Maroon" title="c7">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">a</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_3.html#E7:11"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="borsuk_1.html#D17" title="BORSUK_1:def.17">BORSUK_1:def 17</a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">a</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c6">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">a</font>
<b>by </b><i><a class="txt" href="topalg_3.html#E11:11"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="topalg_3.html#E12:11"><i><font color="Green" title="E12">A7</font></i></a>, <a class="ref" href="tops_2.html#T65" title="TOPS_2:th.65">TOPS_2:65</a></i>
<br/>.= 
<font color="Maroon" title="c8">a</font>
<b>by </b><i><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
;<br/>

<b>thus </b><font color="Maroon" title="c11">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<font color="Maroon" title="c7">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_3.html#E8:11"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="borsuk_1.html#D18" title="BORSUK_1:def.18">BORSUK_1:def 18</a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c6">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font>
<b>by </b><i><a class="txt" href="topalg_3.html#E11:11"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="topalg_3.html#E12:11"><i><font color="Green" title="E12">A7</font></i></a>, <a class="ref" href="tops_2.html#T65" title="TOPS_2:th.65">TOPS_2:65</a></i>
<br/>.= 
<font color="Maroon" title="c9">b</font>
<b>by </b><i><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
;<br/>


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