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<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="pencil_1.html#V3">void</a>  <a href="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="pencil_2.html#NM2">Collineation</a> of <font color="Maroon">S</font>;<br/>



<a NAME="E1:30"/>
<font color="Maroon">f</font> is <a href="funct_2.html#V3">bijective</a>
 
<b>by </b><i/>;<br/>
<a NAME="E2:30"/><b>then </b><i><font color="Green">E32</font></i>: 
( <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">f</font> is <a href="funct_2.html#V2">onto</a> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D4">FUNCT_2:def 4</a></i>;<br/>
<a NAME="E3:30"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E4:30"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a>  as   <a href="pencil_2.html#NM2">Collineation</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>, <font color="Maroon">Y</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/>

<a NAME="E6:30"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 
<b>by </b><i>, , <a class="ref" href="tops_2.html#T68">TOPS_2:68</a></i>;<br/>
<a NAME="E7:30"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, , <a class="ref" href="tops_2.html#T68">TOPS_2:68</a></i>;<br/>
<b>assume </b><a NAME="E8:30"/>
( not <font color="Maroon">X</font> is <a href="realset1.html#V1">trivial</a> &amp; not <font color="Maroon">Y</font> is <a href="realset1.html#V1">trivial</a> &amp; <font color="Maroon">X</font>,<font color="Maroon">Y</font> <a href="pencil_2.html#R1" target="_self">are_joinable</a>  )
 ;<br/>

<b>hence </b><a NAME="E9:30"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">X</font>,<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Y</font> <a href="pencil_2.html#R1" target="_self">are_joinable</a> 
 <b>by </b><i>, , </i>;<br/>


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