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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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>, <font color="Olive" title="b3">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font>  st not <font color="Olive" title="b2">X</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  &amp; not <font color="Olive" title="b3">Y</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  &amp; <font color="Olive" title="b2">X</font>,<font color="Olive" title="b3">Y</font> <a href="pencil_2.html#R1" title="PENCIL_2:pred.1">are_joinable</a>  holds <br/><font color="Olive" title="b1">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Olive" title="b2">X</font>,<font color="Olive" title="b1">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Olive" title="b3">Y</font> <a href="pencil_2.html#R1" title="PENCIL_2:pred.1">are_joinable</a> </span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font>  st not <font color="Olive" title="b1">X</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  &amp; not <font color="Olive" title="b2">Y</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  &amp; <font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">Y</font> <a href="pencil_2.html#R1" title="PENCIL_2:pred.1">are_joinable</a>  holds <br/><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Olive" title="b2">Y</font> <a href="pencil_2.html#R1" title="PENCIL_2:pred.1">are_joinable</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">X</font>, <font color="Maroon" title="c4">Y</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c3">X</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  &amp; not <font color="Maroon" title="c4">Y</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  &amp; <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> <a href="pencil_2.html#R1" title="PENCIL_2:pred.1">are_joinable</a>  implies <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">Y</font> <a href="pencil_2.html#R1" title="PENCIL_2:pred.1">are_joinable</a>  )</span><br/>





<span class="kw">reconsider </span><font color="Maroon" title="c5">g</font> = <font color="Maroon" title="c2">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  as   <a href="pencil_2.html#NM2" title="PENCIL_2:NM.2">Collineation</a> of <font color="Maroon" title="c1">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_2.html#T13" target="_self" title="PENCIL_2:th.13">Th13</a></span>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:30"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
not <font color="Maroon" title="c3">X</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> 
 <span class="kw">and </span><br/><a NAME="E3:30"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
not <font color="Maroon" title="c4">Y</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> 
 <span class="kw">and </span><br/><a NAME="E4:30"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> <a href="pencil_2.html#R1" title="PENCIL_2:pred.1">are_joinable</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">Y</font> <a href="pencil_2.html#R1" title="PENCIL_2:pred.1">are_joinable</a> </span><br/>

<a NAME="E5:30"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
<font color="Maroon" title="c2">f</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_2.html#D4" target="_self" title="PENCIL_2:def.4">Def4</a></span>;<br/>
<a NAME="E6:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A5</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E7:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A6</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="tops_2.html#T68" title="TOPS_2:th.68">TOPS_2:68</a></span>;<br/>
<a NAME="E8:30"/>
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E6:30"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="ref" href="tops_2.html#T68" title="TOPS_2:th.68">TOPS_2:68</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:30"/>
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">Y</font> <a href="pencil_2.html#R1" title="PENCIL_2:pred.1">are_joinable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E3:30"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E4:30"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="pencil_2.html#T20" target="_self" title="PENCIL_2:th.20">Th20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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