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<div class="add">

<b>let </b><font color="Maroon" title="c1">I</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">A</font> being <a href="pencil_1.html#V15" title="PENCIL_1:attr.15">PLS-yielding</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>  st ( for <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> holds  <font color="Olive" title="b1">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <font color="Olive" title="b2">i</font> is <a href="pencil_1.html#V10" title="PENCIL_1:attr.10">strongly_connected</a> ) holds <br/>for <font color="Olive" title="b2">B</font> being   <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Olive" title="b1">A</font><br/> for <font color="Olive" title="b3">f</font> being  <a href="pencil_2.html#NM2" title="PENCIL_2:NM.2">Collineation</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Olive" title="b1">A</font></span>)</span> holds  <font color="Olive" title="b3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b2">B</font> is    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Olive" title="b1">A</font></span><br/><b>let </b><font color="Maroon" title="c2">A</font> be  <a href="pencil_1.html#V15" title="PENCIL_1:attr.15">PLS-yielding</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</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">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> holds  <font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <font color="Olive" title="b1">i</font> is <a href="pencil_1.html#V10" title="PENCIL_1:attr.10">strongly_connected</a> ) implies for <font color="Olive" title="b1">B</font> being   <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font><br/> for <font color="Olive" title="b2">f</font> being  <a href="pencil_2.html#NM2" title="PENCIL_2:NM.2">Collineation</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> holds  <font color="Olive" title="b2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">B</font> is    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> )</span><br/>



<b>assume </b><a NAME="E1:33"/><i><font color="Green" title="E22">A1</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> holds  <font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <font color="Olive" title="b1">i</font> is <a href="pencil_1.html#V10" title="PENCIL_1:attr.10">strongly_connected</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">B</font> being   <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font><br/> for <font color="Olive" title="b2">f</font> being  <a href="pencil_2.html#NM2" title="PENCIL_2:NM.2">Collineation</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> holds  <font color="Olive" title="b2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">B</font> is    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font></span><br/>

<b>let </b><font color="Maroon" title="c3">B</font> be    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</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">f</font> being  <a href="pencil_2.html#NM2" title="PENCIL_2:NM.2">Collineation</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> holds  <font color="Olive" title="b1">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">B</font> is    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font></span><br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="pencil_2.html#NM2" title="PENCIL_2:NM.2">Collineation</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></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">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">B</font> is    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font></span><br/>



<a NAME="E2:33"/>
<font color="Maroon" title="c4">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="E3:33"/><b>then </b><i><font color="Green" title="E23">A2</font></i>: 
( <font color="Maroon" title="c4">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c4">f</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> )
 
;<br/>
<a NAME="E4:33"/><b>then </b><i><font color="Green" title="E24">A3</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">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 <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<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="E5:33"/><i><font color="Green" title="E25">A4</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:33"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">g</font> = <font color="Maroon" title="c4">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 <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> <b>by </b><i><a class="ref" href="pencil_2.html#T13" target="_self" title="PENCIL_2:th.13">Th13</a></i>;<br/>
<a NAME="E7:33"/><i><font color="Green" title="E26">A5</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">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 <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g</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 <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> )
 
<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>consider </b><font color="Maroon" title="c6">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span><b> such that </b><br/><a NAME="E9:33"/><i><font color="Green" title="E27">A6</font></i>: 
( not <font color="Maroon" title="c6">W</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> &amp; <font color="Maroon" title="c6">W</font> is <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a> &amp; <font color="Maroon" title="c6">W</font> is <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a> &amp; <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">Y</font> where <font color="Olive" title="b1">Y</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> : ( not <font color="Olive" title="b1">Y</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> &amp; <font color="Olive" title="b1">Y</font> is <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a> &amp; <font color="Olive" title="b1">Y</font> is <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a> &amp; <font color="Maroon" title="c6">W</font>,<font color="Olive" title="b1">Y</font> <a href="pencil_2.html#R1" title="PENCIL_2:pred.1">are_joinable</a>  ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="#E1:33"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="pencil_2.html#T23" target="_self" title="PENCIL_2:th.23">Th23</a></i>;<br/>
<a NAME="E10:33"/><i><font color="Green" title="E28">A7</font></i>: 
( not <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">W</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> &amp; <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">W</font> is <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a> &amp; <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">W</font> is <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a> )
 
<b>by </b><i><a class="txt" href="#E9:33"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="pencil_2.html#T14" target="_self" title="PENCIL_2:th.14">Th14</a>, <a class="ref" href="pencil_2.html#T16" target="_self" title="PENCIL_2:th.16">Th16</a>, <a class="ref" href="pencil_2.html#T18" target="_self" title="PENCIL_2:th.18">Th18</a></i>;<br/>
<a NAME="E11:33"/>
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">Y</font> where <font color="Olive" title="b1">Y</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> : ( not <font color="Olive" title="b1">Y</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> &amp; <font color="Olive" title="b1">Y</font> is <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a> &amp; <font color="Olive" title="b1">Y</font> is <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a> &amp; <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">W</font>,<font color="Olive" title="b1">Y</font> <a href="pencil_2.html#R1" title="PENCIL_2:pred.1">are_joinable</a>  ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:33_1"/>
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">Y</font> where <font color="Olive" title="b1">Y</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> : ( not <font color="Olive" title="b1">Y</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> &amp; <font color="Olive" title="b1">Y</font> is <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a> &amp; <font color="Olive" title="b1">Y</font> is <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a> &amp; <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">W</font>,<font color="Olive" title="b1">Y</font> <a href="pencil_2.html#R1" title="PENCIL_2:pred.1">are_joinable</a>  ) </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">Y</font> where <font color="Olive" title="b1">Y</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> : ( not <font color="Olive" title="b1">Y</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> &amp; <font color="Olive" title="b1">Y</font> is <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a> &amp; <font color="Olive" title="b1">Y</font> is <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a> &amp; <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">W</font>,<font color="Olive" title="b1">Y</font> <a href="pencil_2.html#R1" title="PENCIL_2:pred.1">are_joinable</a>  ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">B</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">B</font> or <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">Y</font> where <font color="Olive" title="b1">Y</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> : ( not <font color="Olive" title="b1">Y</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> &amp; <font color="Olive" title="b1">Y</font> is <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a> &amp; <font color="Olive" title="b1">Y</font> is <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a> &amp; <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">W</font>,<font color="Olive" title="b1">Y</font> <a href="pencil_2.html#R1" title="PENCIL_2:pred.1">are_joinable</a>  ) </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E1:33_1_1"/>
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">Y</font> where <font color="Olive" title="b1">Y</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> : ( not <font color="Olive" title="b1">Y</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> &amp; <font color="Olive" title="b1">Y</font> is <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a> &amp; <font color="Olive" title="b1">Y</font> is <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a> &amp; <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">W</font>,<font color="Olive" title="b1">Y</font> <a href="pencil_2.html#R1" title="PENCIL_2:pred.1">are_joinable</a>  ) </span>}</span> </span><br/>

<b>then consider </b><font color="Maroon" title="c8">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:33_1_1"/><i><font color="Green" title="E29">A8</font></i>: 
( <font color="Maroon" title="c8">u</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="c4">f</font> &amp; <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c7">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">u</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/>
<b>consider </b><font color="Maroon" title="c9">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:33_1_1"/><i><font color="Green" title="E30">A9</font></i>: 
( <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">Y</font> where <font color="Olive" title="b1">Y</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> : ( not <font color="Olive" title="b1">Y</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> &amp; <font color="Olive" title="b1">Y</font> is <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a> &amp; <font color="Olive" title="b1">Y</font> is <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a> &amp; <font color="Maroon" title="c6">W</font>,<font color="Olive" title="b1">Y</font> <a href="pencil_2.html#R1" title="PENCIL_2:pred.1">are_joinable</a>  ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="#E9:33"><i><font color="Green" title="E27">A6</font></i></a>, <a class="txt" href="#E3:33_1_1"><i><font color="Green" title="E29">A8</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">Y</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span><b> such that </b><br/><a NAME="E7:33_1_1"/><i><font color="Green" title="E31">A10</font></i>: 
( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Y</font> &amp; not <font color="Maroon" title="c10">Y</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> &amp; <font color="Maroon" title="c10">Y</font> is <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a> &amp; <font color="Maroon" title="c10">Y</font> is <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a> &amp; <font color="Maroon" title="c6">W</font>,<font color="Maroon" title="c10">Y</font> <a href="pencil_2.html#R1" title="PENCIL_2:pred.1">are_joinable</a>  )
 <b>by </b><i><a class="txt" href="#E5:33_1_1"><i><font color="Green" title="E30">A9</font></i></a></i>;<br/>
<a NAME="E8:33_1_1"/><i><font color="Green" title="E32">A11</font></i>: 
( not <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">Y</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> &amp; <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">Y</font> is <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a> &amp; <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">Y</font> is <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a> )
 
<b>by </b><i><a class="txt" href="#E7:33_1_1"><i><font color="Green" title="E31">A10</font></i></a>, <a class="ref" href="pencil_2.html#T14" target="_self" title="PENCIL_2:th.14">Th14</a>, <a class="ref" href="pencil_2.html#T16" target="_self" title="PENCIL_2:th.16">Th16</a>, <a class="ref" href="pencil_2.html#T18" target="_self" title="PENCIL_2:th.18">Th18</a></i>;<br/>
<a NAME="E9:33_1_1"/>
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">W</font>,<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">Y</font> <a href="pencil_2.html#R1" title="PENCIL_2:pred.1">are_joinable</a> 
 
<b>by </b><i><a class="txt" href="#E9:33"><i><font color="Green" title="E27">A6</font></i></a>, <a class="txt" href="#E7:33_1_1"><i><font color="Green" title="E31">A10</font></i></a>, <a class="ref" href="pencil_2.html#T20" target="_self" title="PENCIL_2:th.20">Th20</a></i>;<br/>
<a NAME="E10:33_1_1"/><b>then </b><i><font color="Green" title="E33">A12</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">Z</font> where <font color="Olive" title="b1">Z</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> : ( not <font color="Olive" title="b1">Z</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> &amp; <font color="Olive" title="b1">Z</font> is <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a> &amp; <font color="Olive" title="b1">Z</font> is <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a> &amp; <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">W</font>,<font color="Olive" title="b1">Z</font> <a href="pencil_2.html#R1" title="PENCIL_2:pred.1">are_joinable</a>  ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="#E8:33_1_1"><i><font color="Green" title="E32">A11</font></i></a></i>;<br/>
<a NAME="E11:33_1_1"/>
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">Y</font>
 
<b>by </b><i><a class="txt" href="#E3:33_1_1"><i><font color="Green" title="E29">A8</font></i></a>, <a class="txt" href="#E5:33_1_1"><i><font color="Green" title="E30">A9</font></i></a>, <a class="txt" href="#E7:33_1_1"><i><font color="Green" title="E31">A10</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E12:33_1_1"/>
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">Y</font> where <font color="Olive" title="b1">Y</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> : ( not <font color="Olive" title="b1">Y</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> &amp; <font color="Olive" title="b1">Y</font> is <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a> &amp; <font color="Olive" title="b1">Y</font> is <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a> &amp; <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">W</font>,<font color="Olive" title="b1">Y</font> <a href="pencil_2.html#R1" title="PENCIL_2:pred.1">are_joinable</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="#E10:33_1_1"><i><font color="Green" title="E33">A12</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</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>

<b>let </b><font color="Maroon" title="c7">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">Y</font> where <font color="Olive" title="b1">Y</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> : ( not <font color="Olive" title="b1">Y</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> &amp; <font color="Olive" title="b1">Y</font> is <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a> &amp; <font color="Olive" title="b1">Y</font> is <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a> &amp; <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">W</font>,<font color="Olive" title="b1">Y</font> <a href="pencil_2.html#R1" title="PENCIL_2:pred.1">are_joinable</a>  ) </span>}</span>  or <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">B</font> )</span><br/>

<b>assume </b><a NAME="E2:33_1"/>
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">Y</font> where <font color="Olive" title="b1">Y</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> : ( not <font color="Olive" title="b1">Y</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> &amp; <font color="Olive" title="b1">Y</font> is <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a> &amp; <font color="Olive" title="b1">Y</font> is <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a> &amp; <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">W</font>,<font color="Olive" title="b1">Y</font> <a href="pencil_2.html#R1" title="PENCIL_2:pred.1">are_joinable</a>  ) </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="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">B</font></span><br/>

<b>then consider </b><font color="Maroon" title="c8">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:33_1"/><i><font color="Green" title="E29">A13</font></i>: 
( <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">u</font> &amp; <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">Y</font> where <font color="Olive" title="b1">Y</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> : ( not <font color="Olive" title="b1">Y</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> &amp; <font color="Olive" title="b1">Y</font> is <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a> &amp; <font color="Olive" title="b1">Y</font> is <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a> &amp; <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">W</font>,<font color="Olive" title="b1">Y</font> <a href="pencil_2.html#R1" title="PENCIL_2:pred.1">are_joinable</a>  ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">Y</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span><b> such that </b><br/><a NAME="E6:33_1"/><i><font color="Green" title="E30">A14</font></i>: 
( <font color="Maroon" title="c8">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Y</font> &amp; not <font color="Maroon" title="c9">Y</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> &amp; <font color="Maroon" title="c9">Y</font> is <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a> &amp; <font color="Maroon" title="c9">Y</font> is <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a> &amp; <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">W</font>,<font color="Maroon" title="c9">Y</font> <a href="pencil_2.html#R1" title="PENCIL_2:pred.1">are_joinable</a>  )
 <b>by </b><i><a class="txt" href="#E4:33_1"><i><font color="Green" title="E29">A13</font></i></a></i>;<br/>
<a NAME="E7:33_1"/>
( not <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">W</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> &amp; <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">W</font> is <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a> &amp; <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">W</font> is <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a> )
 
<b>by </b><i><a class="txt" href="#E9:33"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="pencil_2.html#T14" target="_self" title="PENCIL_2:th.14">Th14</a>, <a class="ref" href="pencil_2.html#T16" target="_self" title="PENCIL_2:th.16">Th16</a>, <a class="ref" href="pencil_2.html#T18" target="_self" title="PENCIL_2:th.18">Th18</a></i>;<br/>
<a NAME="E8:33_1"/><b>then </b><i><font color="Green" title="E31">A15</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">W</font></span>)</span>,<font color="Maroon" title="c5">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">Y</font> <a href="pencil_2.html#R1" title="PENCIL_2:pred.1">are_joinable</a> 
 
<b>by </b><i><a class="txt" href="#E6:33_1"><i><font color="Green" title="E30">A14</font></i></a>, <a class="ref" href="pencil_2.html#T20" target="_self" title="PENCIL_2:th.20">Th20</a></i>;<br/>
<a NAME="E9:33_1"/><i><font color="Green" title="E32">A16</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">W</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">W</font>
 
<b>by </b><i><a class="txt" href="#E3:33"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="funct_1.html#T152" title="FUNCT_1:th.152">FUNCT_1:152</a></i>;<br/>
<a NAME="E10:33_1"/>
<font color="Maroon" title="c6">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">W</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E7:33"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="funct_1.html#T146" title="FUNCT_1:th.146">FUNCT_1:146</a></i>;<br/>
<a NAME="E11:33_1"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W</font>
 
<b>by </b><i><a class="txt" href="#E9:33_1"><i><font color="Green" title="E32">A16</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E12:33_1"/><b>then </b><i><font color="Green" title="E33">A17</font></i>: 
<font color="Maroon" title="c6">W</font>,<font color="Maroon" title="c5">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">Y</font> <a href="pencil_2.html#R1" title="PENCIL_2:pred.1">are_joinable</a> 
 
<b>by </b><i><a class="txt" href="#E3:33"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="#E5:33"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="#E8:33_1"><i><font color="Green" title="E31">A15</font></i></a>, <a class="ref" href="tops_2.html#T68" title="TOPS_2:th.68">TOPS_2:68</a></i>;<br/>
<a NAME="E13:33_1"/>
( not <font color="Maroon" title="c5">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">Y</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> &amp; <font color="Maroon" title="c5">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">Y</font> is <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a> &amp; <font color="Maroon" title="c5">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">Y</font> is <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a> )
 
<b>by </b><i><a class="txt" href="#E6:33_1"><i><font color="Green" title="E30">A14</font></i></a>, <a class="ref" href="pencil_2.html#T14" target="_self" title="PENCIL_2:th.14">Th14</a>, <a class="ref" href="pencil_2.html#T16" target="_self" title="PENCIL_2:th.16">Th16</a>, <a class="ref" href="pencil_2.html#T18" target="_self" title="PENCIL_2:th.18">Th18</a></i>;<br/>
<a NAME="E14:33_1"/><b>then </b><i><font color="Green" title="E34">A18</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">Z</font> where <font color="Olive" title="b1">Z</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> : ( not <font color="Olive" title="b1">Z</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> &amp; <font color="Olive" title="b1">Z</font> is <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a> &amp; <font color="Olive" title="b1">Z</font> is <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a> &amp; <font color="Maroon" title="c6">W</font>,<font color="Olive" title="b1">Z</font> <a href="pencil_2.html#R1" title="PENCIL_2:pred.1">are_joinable</a>  ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="#E12:33_1"><i><font color="Green" title="E33">A17</font></i></a></i>;<br/>
<a NAME="E15:33_1"/>
<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">Y</font>
 
<b>by </b><i><a class="txt" href="#E7:33"><i><font color="Green" title="E26">A5</font></i></a>, <a class="txt" href="#E4:33_1"><i><font color="Green" title="E29">A13</font></i></a>, <a class="txt" href="#E6:33_1"><i><font color="Green" title="E30">A14</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="E16:33_1"/><b>then </b><i><font color="Green" title="E35">A19</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 
<b>by </b><i><a class="txt" href="#E9:33"><i><font color="Green" title="E27">A6</font></i></a>, <a class="txt" href="#E14:33_1"><i><font color="Green" title="E34">A18</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E17:33_1"/>
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">o</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:33"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="#E4:33"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="#E4:33_1"><i><font color="Green" title="E29">A13</font></i></a>, <a class="txt" href="#E6:33_1"><i><font color="Green" title="E30">A14</font></i></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></i>;<br/>
<a NAME="E18:33_1"/><b>then </b>
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">o</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:33"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="#E4:33"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></i>;<br/>
<b>hence </b><a NAME="E19:33_1"/>
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="#E7:33"><i><font color="Green" title="E26">A5</font></i></a>, <a class="txt" href="#E16:33_1"><i><font color="Green" title="E35">A19</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</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>
<b>hence </b><a NAME="E12:33"/>
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">B</font> is    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="#E1:33"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="#E10:33"><i><font color="Green" title="E28">A7</font></i></a>, <a class="ref" href="pencil_2.html#T23" target="_self" title="PENCIL_2:th.23">Th23</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>
