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<div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#NM1" title="PBOOLE:NM.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="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <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> iff  ex <font color="Olive" title="b3">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#K5" title="PENCIL_1:func.5">Segre_Product</a> <font color="Olive" title="b1">A</font></span>)</span> st <br/>(  not <font color="Olive" title="b3">W</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  &amp; <font color="Olive" title="b3">W</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Olive" title="b3">W</font> is  <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a>  &amp; <font color="Olive" title="b2">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="b4">Y</font> where <font color="Olive" title="b4">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#K5" title="PENCIL_1:func.5">Segre_Product</a> <font color="Olive" title="b1">A</font></span>)</span> : (  not <font color="Olive" title="b4">Y</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  &amp; <font color="Olive" title="b4">Y</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Olive" title="b4">Y</font> is  <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a>  &amp; <font color="Olive" title="b3">W</font>,<font color="Olive" title="b4">Y</font> <a href="pencil_2.html#R1" title="PENCIL_2:pred.1">are_joinable</a>  ) </span> } </span>  ) )</span><br/><span class="kw">let </span><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#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <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> iff  ex <font color="Olive" title="b2">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#K5" title="PENCIL_1:func.5">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> st <br/>(  not <font color="Olive" title="b2">W</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  &amp; <font color="Olive" title="b2">W</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Olive" title="b2">W</font> is  <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a>  &amp; <font color="Olive" title="b1">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="b3">Y</font> where <font color="Olive" title="b3">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#K5" title="PENCIL_1:func.5">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> : (  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="b3">Y</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Olive" title="b3">Y</font> is  <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a>  &amp; <font color="Olive" title="b2">W</font>,<font color="Olive" title="b3">Y</font> <a href="pencil_2.html#R1" title="PENCIL_2:pred.1">are_joinable</a>  ) </span> } </span>  ) ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:32"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
 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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <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> iff  ex <font color="Olive" title="b2">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#K5" title="PENCIL_1:func.5">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> st <br/>(  not <font color="Olive" title="b2">W</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  &amp; <font color="Olive" title="b2">W</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Olive" title="b2">W</font> is  <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a>  &amp; <font color="Olive" title="b1">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="b3">Y</font> where <font color="Olive" title="b3">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#K5" title="PENCIL_1:func.5">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> : (  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="b3">Y</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Olive" title="b3">Y</font> is  <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a>  &amp; <font color="Olive" title="b2">W</font>,<font color="Olive" title="b3">Y</font> <a href="pencil_2.html#R1" title="PENCIL_2:pred.1">are_joinable</a>  ) </span> } </span>  ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">B</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> iff  ex <font color="Olive" title="b1">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#K5" title="PENCIL_1:func.5">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> st <br/>(  not <font color="Olive" title="b1">W</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  &amp; <font color="Olive" title="b1">W</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Olive" title="b1">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="b2">Y</font> where <font color="Olive" title="b2">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#K5" title="PENCIL_1:func.5">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> : (  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="b2">Y</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Olive" title="b2">Y</font> is  <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a>  &amp; <font color="Olive" title="b1">W</font>,<font color="Olive" title="b2">Y</font> <a href="pencil_2.html#R1" title="PENCIL_2:pred.1">are_joinable</a>  ) </span> } </span>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E2:32"/>
( <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> implies  ex <font color="Olive" title="b1">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#K5" title="PENCIL_1:func.5">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> st <br/>(  not <font color="Olive" title="b1">W</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  &amp; <font color="Olive" title="b1">W</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Olive" title="b1">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="b2">Y</font> where <font color="Olive" title="b2">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#K5" title="PENCIL_1:func.5">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> : (  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="b2">Y</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Olive" title="b2">Y</font> is  <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a>  &amp; <font color="Olive" title="b1">W</font>,<font color="Olive" title="b2">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">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#K5" title="PENCIL_1:func.5">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> st <br/>(  not <font color="Olive" title="b1">W</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  &amp; <font color="Olive" title="b1">W</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Olive" title="b1">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="b2">Y</font> where <font color="Olive" title="b2">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#K5" title="PENCIL_1:func.5">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> : (  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="b2">Y</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Olive" title="b2">Y</font> is  <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a>  &amp; <font color="Olive" title="b1">W</font>,<font color="Olive" title="b2">Y</font> <a href="pencil_2.html#R1" title="PENCIL_2:pred.1">are_joinable</a>  ) </span> } </span>  ) implies <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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:32_1"/>
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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#K5" title="PENCIL_1:func.5">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> st <br/>(  not <font color="Olive" title="b1">W</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  &amp; <font color="Olive" title="b1">W</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Olive" title="b1">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="b2">Y</font> where <font color="Olive" title="b2">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#K5" title="PENCIL_1:func.5">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> : (  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="b2">Y</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Olive" title="b2">Y</font> is  <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a>  &amp; <font color="Olive" title="b1">W</font>,<font color="Olive" title="b2">Y</font> <a href="pencil_2.html#R1" title="PENCIL_2:pred.1">are_joinable</a>  ) </span> } </span>  )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">BB</font> = <font color="Maroon" title="c3">B</font> as    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">L</font> being   non  <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a>   <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>   <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font><span class="kw"> such that </span><br/><a NAME="E4:32_1"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
<font color="Maroon" title="c4">BB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c5">L</font>
 <span class="kw">and </span><br/><a NAME="E5:32_1"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
<font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">L</font></span>)</span> <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"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">L</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_2.html#D2" target="_self" title="PENCIL_2:def.2">Def2</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">K1</font> =  the   <a href="pencil_1.html#NM1" title="PENCIL_1:NM.1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">L</font></span>)</span></span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">V</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:32_1"/><span class="lab"><font color="Green" title="E24">A4</font></span>: 
<font color="Maroon" title="c7">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c5">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E8:32_1"/>
 the   <a href="pencil_1.html#NM1" title="PENCIL_1:NM.1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">L</font></span>)</span></span>)</span> <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 <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">L</font></span>)</span></span>)</span>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">K</font> =  the   <a href="pencil_1.html#NM1" title="PENCIL_1:NM.1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">L</font></span>)</span></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">L</font></span>)</span></span>)</span> ;<br/>
<a NAME="E10:32_1"/><span class="lab"><font color="Green" title="E25">A5</font></span>: 
 ex <font color="Olive" title="b1">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">V</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">L</font> &amp; (  for <font color="Olive" title="b2">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">L</font> holds <br/><font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:32_1"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></span>;<br/>
<a NAME="E11:32_1"/><span class="lab"><font color="Green" title="E26">A6</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">V</font> = <font color="Maroon" title="c7">V</font> as   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:32_1"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a>, <a class="ref" href="relat_1.html#D18" title="RELAT_1:def.18">RELAT_1:def 18</a></span>;<br/>
<a NAME="E13:32_1"/>
 for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Maroon" title="c9">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> implies <font color="Maroon" title="c9">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_1_1"/><span class="lab"><font color="Green" title="E27">A7</font></span>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span><br/>

<a NAME="E2:32_1_1"/>
<font color="Maroon" title="c5">L</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></span>;<br/>
<a NAME="E3:32_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A8</font></span>: 
<font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="ref" href="pboole.html#D2" title="PBOOLE:def.2">PBOOLE:def 2</a></span>;<br/>
<a NAME="E4:32_1_1"/>
<font color="Maroon" title="c9">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:32_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="txt" href="#E10:32_1"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="txt" href="#E1:32_1_1"><span class="lab"><font color="Green" title="E27">A7</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:32_1_1"/>
<font color="Maroon" title="c9">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_1"><span class="lab"><font color="Green" title="E28">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">V</font> = <font color="Maroon" title="c9">V</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">Element</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D14" title="PBOOLE:def.14">PBOOLE:def 14</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">VV</font> = <span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">V</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> as    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#T11" title="PENCIL_1:th.11">PENCIL_1:11</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">X</font> = <font color="Maroon" title="c11">VV</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">L</font></span>)</span>,<font color="Maroon" title="c8">K</font>) as    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_2.html#T7" target="_self" title="PENCIL_2:th.7">Th7</a></span>;<br/>
<a NAME="E17:32_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> <font color="Maroon" title="c8">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#D6" title="PENCIL_1:def.6">PENCIL_1:def 6</a></span>;<br/>
<a NAME="E18:32_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A9</font></span>: 
 not <font color="Maroon" title="c8">K</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#T4" title="PENCIL_1:th.4">PENCIL_1:4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">X</font> = <font color="Maroon" title="c12">X</font> as   non  <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a>   <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>   <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#T9" title="PENCIL_1:th.9">PENCIL_1:9</a>, <a class="ref" href="pencil_1.html#T10" title="PENCIL_1:th.10">PENCIL_1:10</a></span>;<br/>
<a NAME="E20:32_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">VV</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E21:32_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A10</font></span>: 
<font color="Maroon" title="c13">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T31" title="FUNCT_7:th.31">FUNCT_7:31</a></span>;<br/>
<a NAME="E22:32_1"/><span class="kw">then </span>
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c13">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:32_1"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="ref" href="pencil_1.html#D21" title="PENCIL_1:def.21">PENCIL_1:def 21</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">pX</font> =  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c13">X</font> as   <a href="pencil_1.html#NM1" title="PENCIL_1:NM.1">Block</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K5" title="PENCIL_1:func.5">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:32_1"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="ref" href="pencil_1.html#T24" title="PENCIL_1:th.24">PENCIL_1:24</a></span>;<br/>
<a NAME="E24:32_1"/><span class="lab"><font color="Green" title="E29">A11</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Maroon" title="c13">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> implies <font color="Maroon" title="c13">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_1_2"/><span class="lab"><font color="Green" title="E30">A12</font></span>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_1_2_1"/>
( <font color="Maroon" title="c15">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">L</font> or <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">L</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_2_1_1"/>
<font color="Maroon" title="c15">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font></span><br/><div class="add"><a NAME="E2:32_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">VV</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></span>;<br/><a NAME="E3:32_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A13</font></span>: 
<font color="Maroon" title="c13">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_2"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="ref" href="pzfmisc1.html#D1" title="PZFMISC1:def.1">PZFMISC1:def 1</a></span>;<br/><a NAME="E4:32_1_2_1_1"/>
<font color="Maroon" title="c10">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:32_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="txt" href="#E10:32_1"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="txt" href="#E1:32_1_2"><span class="lab"><font color="Green" title="E30">A12</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:32_1_2_1_1"/>
<font color="Maroon" title="c13">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_2_1_1"><span class="lab"><font color="Green" title="E31">A13</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_2_1_2"/>
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:32_1_2_1_2"/>
<font color="Maroon" title="c13">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_1"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="txt" href="#E21:32_1"><span class="lab"><font color="Green" title="E28">A10</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E25:32_1"/>
<font color="Maroon" title="c14">pX</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 <span class="p1">(<span class="default"><a href="pencil_1.html#K5" title="PENCIL_1:func.5">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">psX</font> = <font color="Maroon" title="c14">pX</font> as   <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#K5" title="PENCIL_1:func.5">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c15">psX</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c15">psX</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  &amp; <font color="Maroon" title="c15">psX</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Maroon" title="c15">psX</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#K5" title="PENCIL_1:func.5">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="c15">psX</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/>

<span class="kw">thus </span><a NAME="E27:32_1"/><span class="lab"><font color="Green" title="E30">A14</font></span>: 
(  not <font color="Maroon" title="c15">psX</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  &amp; <font color="Maroon" title="c15">psX</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Maroon" title="c15">psX</font> is  <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#T20" title="PENCIL_1:th.20">PENCIL_1:20</a>, <a class="ref" href="pencil_1.html#T21" title="PENCIL_1:th.21">PENCIL_1:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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#K5" title="PENCIL_1:func.5">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="c15">psX</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/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">Z</font> =  <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#K5" title="PENCIL_1:func.5">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="c15">psX</font>,<font color="Olive" title="b1">Y</font> <a href="pencil_2.html#R1" title="PENCIL_2:pred.1">are_joinable</a>  ) </span> } </span>  as    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="pencil_2.html#T22" target="_self" title="PENCIL_2:th.22">Th22</a></span>;<br/>
<a NAME="E29:32_1"/>
<font color="Maroon" title="c15">psX</font>,<font color="Maroon" title="c15">psX</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="#E27:32_1"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="ref" href="pencil_2.html#T10" target="_self" title="PENCIL_2:th.10">Th10</a></span>;<br/>
<a NAME="E30:32_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A15</font></span>: 
<font color="Maroon" title="c15">psX</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#K5" title="PENCIL_1:func.5">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="c15">psX</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E27:32_1"><span class="lab"><font color="Green" title="E30">A14</font></span></a></span>;<br/>
<a NAME="E31:32_1"/><span class="lab"><font color="Green" title="E32">A16</font></span>: 
<font color="Maroon" title="c15">psX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">Z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">psX</font> or <font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Z</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_1_3"/>
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">psX</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Z</font></span><br/>

<span class="kw">hence </span><a NAME="E2:32_1_3"/>
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:32_1"><span class="lab"><font color="Green" title="E31">A15</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E32:32_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E33:32_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">psX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="txt" href="#E11:32_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="txt" href="#E24:32_1"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="ref" href="card_3.html#T27" title="CARD_3:th.27">CARD_3:27</a></span>;<br/>
<a NAME="E34:32_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">psX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c16">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:32_1"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>;<br/>
<a NAME="E35:32_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A17</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c15">psX</font> <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="c3">B</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c16">Z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a></span>;<br/>
<a NAME="E36:32_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> <font color="Maroon" title="c14">pX</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#D6" title="PENCIL_1:def.6">PENCIL_1:def 6</a></span>;<br/>
<a NAME="E37:32_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">BB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:32_1"><span class="lab"><font color="Green" title="E33">A17</font></span></a>, <a class="ref" href="pencil_2.html#T8" target="_self" title="PENCIL_2:th.8">Th8</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E38:32_1"/>
<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#K5" title="PENCIL_1:func.5">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="c15">psX</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">given </span><font color="Maroon" title="c4">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#K5" title="PENCIL_1:func.5">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span><span class="kw"> such that </span><a NAME="E4:32"/><span class="lab"><font color="Green" title="E22">A18</font></span>: 
(  not <font color="Maroon" title="c4">W</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  &amp; <font color="Maroon" title="c4">W</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Maroon" title="c4">W</font> is  <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a>  )
 <span class="kw">and </span><br/><a NAME="E5:32"/><span class="lab"><font color="Green" title="E23">A19</font></span>: 
<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#K5" title="PENCIL_1:func.5">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">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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/>

<span class="kw">thus </span><a NAME="E6:32"/>
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E4:32"><span class="lab"><font color="Green" title="E22">A18</font></span></a>, <a class="txt" href="#E5:32"><span class="lab"><font color="Green" title="E23">A19</font></span></a>, <a class="ref" href="pencil_2.html#T22" target="_self" title="PENCIL_2:th.22">Th22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
