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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> non <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>  <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="pencil_2.html#NM2" title="PENCIL_2:NM.2">Collineation</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b2">X</font> is <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a> holds <br/><font color="Olive" title="b1">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b2">X</font> is <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">X</font> is <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a> holds <br/><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">X</font> is <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">X</font> is <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a> implies <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="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a> )</span><br/>





<b>assume </b><a NAME="E1:27"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c3">X</font> is <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a></span><br/>

<b>thus </b><a NAME="E2:27"/>
<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="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">l</font> be   <a href="pencil_1.html#NM1" title="PENCIL_1:NM.1">Block</a> of <font color="Maroon" title="c1">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="pencil_1.html#D2" title="PENCIL_1:def.2">PENCIL_1:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not 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="c4">l</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<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></span>)</span> or <font color="Maroon" title="c4">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> )</span><br/>

<a NAME="E1:27_1"/><i><font color="Green" title="E19">A2</font></i>: 
<font color="Maroon" title="c4">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">S</font>
 
;<br/>
<a NAME="E2:27_1"/>
<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="E3:27_1"/><b>then </b>
<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="E4:27_1"/><b>then </b><i><font color="Green" title="E20">A3</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> 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#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<b>assume </b><a NAME="E5:27_1"/>
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="c4">l</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<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></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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></span><br/>

<b>then consider </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:27_1"/><i><font color="Green" title="E21">A4</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">l</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">l</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</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="E8:27_1"/><i><font color="Green" title="E22">A5</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">l</font> &amp; <font color="Maroon" title="c5">a</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#E7:27_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:27_1"/><i><font color="Green" title="E23">A6</font></i>: 
( <font color="Maroon" title="c7">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c7">A</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">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E11:27_1"/><i><font color="Green" title="E24">A7</font></i>: 
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">l</font> &amp; <font color="Maroon" title="c6">b</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#E7:27_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:27_1"/><i><font color="Green" title="E25">A8</font></i>: 
( <font color="Maroon" title="c8">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c8">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">B</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E14:27_1"/>
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="pencil_2.html#K3" title="PENCIL_2:func.3">"</a> <font color="Maroon" title="c4">l</font>
 
<b>by </b><i><a class="txt" href="pencil_2.html#E8:27_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="pencil_2.html#E10:27_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E15:27_1"/><b>then </b><i><font color="Green" title="E26">A9</font></i>: 
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="pencil_2.html#K3" title="PENCIL_2:func.3">"</a> <font color="Maroon" title="c4">l</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="txt" href="pencil_2.html#E10:27_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E16:27_1"/>
<font color="Maroon" title="c8">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="pencil_2.html#K3" title="PENCIL_2:func.3">"</a> <font color="Maroon" title="c4">l</font>
 
<b>by </b><i><a class="txt" href="pencil_2.html#E11:27_1"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="pencil_2.html#E13:27_1"><i><font color="Green" title="E25">A8</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E17:27_1"/><b>then </b>
<font color="Maroon" title="c8">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="pencil_2.html#K3" title="PENCIL_2:func.3">"</a> <font color="Maroon" title="c4">l</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="txt" href="pencil_2.html#E13:27_1"><i><font color="Green" title="E25">A8</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E18:27_1"/><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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="pencil_2.html#K3" title="PENCIL_2:func.3">"</a> <font color="Maroon" title="c4">l</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_2.html#E7:27_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="pencil_2.html#E10:27_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="pencil_2.html#E13:27_1"><i><font color="Green" title="E25">A8</font></i></a>, <a class="txt" href="pencil_2.html#E15:27_1"><i><font color="Green" title="E26">A9</font></i></a>, <a class="ref" href="pencil_1.html#T2" title="PENCIL_1:th.2">PENCIL_1:2</a></i>;<br/>
<a NAME="E19:27_1"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="pencil_2.html#K3" title="PENCIL_2:func.3">"</a> <font color="Maroon" title="c4">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="txt" href="pencil_2.html#E1:27"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="pencil_1.html#D2" title="PENCIL_1:def.2">PENCIL_1:def 2</a></i>;<br/>
<a NAME="E20:27_1"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="pencil_2.html#K2" title="PENCIL_2:func.2">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="pencil_2.html#K3" title="PENCIL_2:func.3">"</a> <font color="Maroon" title="c4">l</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a></i>;<br/>
<b>hence </b><a NAME="E21:27_1"/>
<font color="Maroon" title="c4">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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#E1:27_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="pencil_2.html#E4:27_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="funct_1.html#T147" title="FUNCT_1:th.147">FUNCT_1:147</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>


</div>
