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<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">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/><b>let </b><font color="Maroon" title="c3">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font>;<br/>





<a NAME="E1:23"/>
<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="E2:23"/><b>then </b><i><font color="Green" title="E16">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>
 
<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="E3:23"/><i><font color="Green" title="E17">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</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="c1">S</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>assume </b><a NAME="E4:23"/>
not <font color="Maroon" title="c3">X</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 ;<br/>

<a NAME="E5:23"/><b>then </b>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="ref" href="pencil_1.html#T4" title="PENCIL_1:th.4">PENCIL_1:4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:23"/><i><font color="Green" title="E18">A3</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">y</font> )
 <b>by </b><i><a class="ref" href="pencil_1.html#T2" title="PENCIL_1:th.2">PENCIL_1:2</a></i>;<br/>
<a NAME="E8:23"/><i><font color="Green" title="E19">A4</font></i>: 
( <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">X</font> )
 
<b>by </b><i><a class="txt" href="pencil_2.html#E3:23"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="pencil_2.html#E7:23"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E9:23"/>
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font>
 
<b>by </b><i><a class="txt" href="pencil_2.html#E2:23"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="pencil_2.html#E3:23"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="pencil_2.html#E7:23"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E10:23"/><b>then </b>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_2.html#E8:23"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="pencil_1.html#T2" title="PENCIL_1:th.2">PENCIL_1:2</a></i>;<br/>
<b>hence </b><a NAME="E11:23"/>
not <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">X</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 <b>by </b><i><a class="ref" href="pencil_1.html#T4" title="PENCIL_1:th.4">PENCIL_1:4</a></i>;<br/>


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