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<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> ;<br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="pencil_2.html#NM2" title="PENCIL_2:NM.2">Collineation</a> of <font color="Maroon" title="c1">S</font>;<br/>



<a NAME="E1:22"/>
<font color="Maroon" title="c2">f</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>
 
<b>by </b><i><a class="ref" href="pencil_2.html#D4" target="_self" title="PENCIL_2:def.4">Def4</a></i>;<br/>
<a NAME="E2:22"/><b>then </b><i><font color="Green" title="E15">A1</font></i>: 
( <font color="Maroon" title="c2">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c2">f</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D4" title="FUNCT_2:def.4">FUNCT_2:def 4</a></i>;<br/>
<a NAME="E3:22"/><b>then </b><i><font color="Green" title="E16">A2</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</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="c1">S</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E4:22"/>
<font color="Maroon" title="c2">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="pencil_2.html#V1" title="PENCIL_2:attr.1">isomorphic</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:22_1"/>
( <font color="Maroon" title="c2">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> &amp; <font color="Maroon" title="c2">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="t_0topsp.html#V1" title="T_0TOPSP:attr.1">open</a> )
 <b>by </b><i><a class="ref" href="pencil_2.html#D4" target="_self" title="PENCIL_2:def.4">Def4</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pencil_2.html#D4" target="_self" title="PENCIL_2:def.4">PENCIL_2:def 4</a><br/>

<a NAME="E2:22_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="pencil_2.html#E2:22"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="pencil_2.html#E3:22"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="tops_2.html#T64" title="TOPS_2:th.64">TOPS_2:64</a></i>;<br/>
<b>hence </b><a NAME="E3:22_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="t_0topsp.html#V1" title="T_0TOPSP:attr.1">open</a> )
 <b>by </b><i><a class="ref" href="pencil_2.html#D4" target="_self" title="PENCIL_2:def.4">Def4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:22"/>
<font color="Maroon" title="c2">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is   <a href="pencil_2.html#NM2" title="PENCIL_2:NM.2">Collineation</a> of <font color="Maroon" title="c1">S</font>
 ;<br/>


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