<?xml version="1.0"?>
<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><br/>  for <font color="Olive" title="b2">S</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> holds <br/> ( (  not <font color="Olive" title="b2">S</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  &amp; <font color="Olive" title="b2">S</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Olive" title="b2">S</font> is  <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a>  ) iff  ex <font color="Olive" title="b3">B</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="Olive" title="b1">A</font> st <br/>( <font color="Olive" title="b2">S</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="Olive" title="b3">B</font> &amp; (  for <font color="Olive" title="b4">C</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">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="Olive" title="b3">B</font></span>)</span></span>)</span>  st <font color="Olive" title="b4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">B</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="Olive" title="b3">B</font></span>)</span> holds <br/>( <font color="Olive" title="b4">C</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Olive" title="b4">C</font> is  <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a>  ) ) ) )</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">S</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> holds <br/> ( (  not <font color="Olive" title="b1">S</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  &amp; <font color="Olive" title="b1">S</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Olive" title="b1">S</font> is  <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a>  ) iff  ex <font color="Olive" title="b2">B</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> st <br/>( <font color="Olive" title="b1">S</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="Olive" title="b2">B</font> &amp; (  for <font color="Olive" title="b3">C</font> being   <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="Olive" title="b2">B</font></span>)</span></span>)</span>  st <font color="Olive" title="b3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B</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="Olive" title="b2">B</font></span>)</span> holds <br/>( <font color="Olive" title="b3">C</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Olive" title="b3">C</font> is  <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a>  ) ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">S</font> be   <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>; <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="c3">S</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  &amp; <font color="Maroon" title="c3">S</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Maroon" title="c3">S</font> is  <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a>  ) iff  ex <font color="Olive" title="b1">B</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> st <br/>( <font color="Maroon" title="c3">S</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="Olive" title="b1">B</font> &amp; (  for <font color="Olive" title="b2">C</font> being   <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="Olive" title="b1">B</font></span>)</span></span>)</span>  st <font color="Olive" title="b2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</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="Olive" title="b1">B</font></span>)</span> holds <br/>( <font color="Olive" title="b2">C</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Olive" title="b2">C</font> is  <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a>  ) ) ) )</span><br/>





<span class="kw">thus </span><a NAME="E1:64"/>
(  not <font color="Maroon" title="c3">S</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  &amp; <font color="Maroon" title="c3">S</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Maroon" title="c3">S</font> is  <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a>  implies  ex <font color="Olive" title="b1">B</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> st <br/>( <font color="Maroon" title="c3">S</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="Olive" title="b1">B</font> &amp; (  for <font color="Olive" title="b2">C</font> being   <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="Olive" title="b1">B</font></span>)</span></span>)</span>  st <font color="Olive" title="b2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</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="Olive" title="b1">B</font></span>)</span> holds <br/>( <font color="Olive" title="b2">C</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Olive" title="b2">C</font> is  <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a>  ) ) ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">B</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> st <br/>( <font color="Maroon" title="c3">S</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="Olive" title="b1">B</font> &amp; (  for <font color="Olive" title="b2">C</font> being   <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="Olive" title="b1">B</font></span>)</span></span>)</span>  st <font color="Olive" title="b2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</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="Olive" title="b1">B</font></span>)</span> holds <br/>( <font color="Olive" title="b2">C</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Olive" title="b2">C</font> is  <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a>  ) ) ) implies (  not <font color="Maroon" title="c3">S</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  &amp; <font color="Maroon" title="c3">S</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Maroon" title="c3">S</font> is  <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a>  ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:64_1"/><span class="lab"><font color="Green" title="E42">A1</font></span>: 
(  not <font color="Maroon" title="c3">S</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  &amp; <font color="Maroon" title="c3">S</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Maroon" title="c3">S</font> is  <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">B</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> st <br/>( <font color="Maroon" title="c3">S</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="Olive" title="b1">B</font> &amp; (  for <font color="Olive" title="b2">C</font> being   <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="Olive" title="b1">B</font></span>)</span></span>)</span>  st <font color="Olive" title="b2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</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="Olive" title="b1">B</font></span>)</span> holds <br/>( <font color="Olive" title="b2">C</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Olive" title="b2">C</font> is  <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a>  ) ) )</span><br/>

<a NAME="E2:64_1"/><span class="kw">then </span>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c3">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#T4" target="_self" title="PENCIL_1:th.4">Th4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:64_1"/><span class="lab"><font color="Green" title="E43">A2</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font>
 <span class="kw">and </span><br/><a NAME="E5:64_1"/><span class="lab"><font color="Green" title="E44">A3</font></span>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font>
 <span class="kw">and </span><br/><a NAME="E6:64_1"/><span class="lab"><font color="Green" title="E45">A4</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#T2" target="_self" title="PENCIL_1:th.2">Th2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">a</font> = <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c7">b</font> = <font color="Maroon" title="c5">b</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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="#E4:64_1"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E5:64_1"><span class="lab"><font color="Green" title="E44">A3</font></span></a></span>;<br/>
<a NAME="E8:64_1"/>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="txt" href="#E4:64_1"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E5:64_1"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="ref" href="pencil_1.html#D3" target="_self" title="PENCIL_1:def.3">Def3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">C</font> being   <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"> such that </span><br/><a NAME="E10:64_1"/><span class="lab"><font color="Green" title="E46">A5</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:64_1"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="ref" href="pencil_1.html#D1" target="_self" title="PENCIL_1:def.1">Def1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">CC</font> being   <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="E12:64_1"/><span class="lab"><font color="Green" title="E47">A6</font></span>: 
<font color="Maroon" title="c8">C</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="c9">CC</font>
 <span class="kw">and </span><br/><a NAME="E13:64_1"/>
 ex <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> st <font color="Maroon" title="c9">CC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is   <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> <font color="Olive" title="b1">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#D22" target="_self" title="PENCIL_1:def.22">Def22</a></span>;<br/>
<a NAME="E14:64_1"/>
( <font color="Maroon" title="c6">a</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="c9">CC</font> &amp; <font color="Maroon" title="c7">b</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="c9">CC</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:64_1"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="txt" href="#E12:64_1"><span class="lab"><font color="Green" title="E47">A6</font></span></a>, <a class="ref" href="zfmisc_1.html#T32" title="ZFMISC_1:th.32">ZFMISC_1:32</a></span>;<br/>
<a NAME="E15:64_1"/><span class="kw">then </span>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c9">CC</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:64_1"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="ref" href="pencil_1.html#T2" target="_self" title="PENCIL_1:th.2">Th2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">CC</font> = <font color="Maroon" title="c9">CC</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#T13" target="_self" title="PENCIL_1:th.13">Th13</a></span>;<br/>
<a NAME="E17:64_1"/><span class="lab"><font color="Green" title="E48">A7</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">CC</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">reconsider </span><font color="Maroon" title="c11">a1</font> = <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c12">b1</font> = <font color="Maroon" title="c7">b</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="ref" href="pencil_1.html#T14" target="_self" title="PENCIL_1:th.14">Th14</a></span>;<br/>
<a NAME="E19:64_1"/><span class="lab"><font color="Green" title="E49">A8</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">a1</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="E20:64_1"/><span class="lab"><font color="Green" title="E50">A9</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">b1</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/>
<div><span class="lab"><font color="Green" title="E51">A10</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:64_1_1"/><span class="lab"><font color="Green" title="E52">A11</font></span>: 
<font color="Maroon" title="c11">a1</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="c10">CC</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">b1</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="c10">CC</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:64_1_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="c11">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">b1</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="64_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">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="c13">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="c11">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_1_1_1"/>
<font color="Maroon" title="c13">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="c11">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_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="E1:64_1_1_1_1"/>
( <font color="Maroon" title="c13">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="c10">CC</font> or <font color="Maroon" title="c13">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="c10">CC</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:64_1_1_1_1_1"/>
<font color="Maroon" title="c13">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="c10">CC</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="c11">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:64_1_1_1_1_1"/>
<font color="Maroon" title="c11">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1"><span class="lab"><font color="Green" title="E52">A11</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:64_1_1_1_1_2"/><span class="lab"><font color="Green" title="E53">A12</font></span>: 
<font color="Maroon" title="c13">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="c10">CC</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="c11">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font></span><br/><div class="add"><a NAME="E2:64_1_1_1_1_2"/>
( <font color="Maroon" title="c11">a1</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="c10">CC</font> &amp; <font color="Maroon" title="c12">b1</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="c10">CC</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:64_1"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="txt" href="#E12:64_1"><span class="lab"><font color="Green" title="E47">A6</font></span></a>, <a class="ref" href="zfmisc_1.html#T32" title="ZFMISC_1:th.32">ZFMISC_1:32</a></span>;<br/><span class="kw">hence </span><a NAME="E3:64_1_1_1_1_2"/>
<font color="Maroon" title="c11">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1_1_1_2"><span class="lab"><font color="Green" title="E53">A12</font></span></a>, <a class="ref" href="pencil_1.html#T23" target="_self" title="PENCIL_1:th.23">Th23</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><span class="kw">hence </span><a NAME="E3:64_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:64_1"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="txt" href="#E19:64_1"><span class="lab"><font color="Green" title="E49">A8</font></span></a>, <a class="txt" href="#E20:64_1"><span class="lab"><font color="Green" title="E50">A9</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</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="E22:64_1"/><span class="lab"><font color="Green" title="E52">A13</font></span>: 
 for <font color="Olive" title="b1">f</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font> holds <br/> 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>  st <font color="Olive" title="b2">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="c10">CC</font> holds <br/><font color="Olive" title="b1">f</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="c10">CC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">f</font> be   <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"> ( <font color="Maroon" title="c13">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font> implies  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>  st <font color="Olive" title="b1">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="c10">CC</font> holds <br/><font color="Maroon" title="c13">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">CC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_1_2"/><span class="lab"><font color="Green" title="E53">A14</font></span>: 
<font color="Maroon" title="c13">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</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>  st <font color="Olive" title="b1">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="c10">CC</font> holds <br/><font color="Maroon" title="c13">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">CC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">f1</font> = <font color="Maroon" title="c13">f</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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">let </span><font color="Maroon" title="c15">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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"> ( <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="c10">CC</font> implies <font color="Maroon" title="c13">f</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="c10">CC</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="E3:64_1_2"/><span class="lab"><font color="Green" title="E54">A15</font></span>: 
<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="c10">CC</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">f</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="c10">CC</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="hs2(this)" href="javascript:()" title="64_1_2_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:64_1_2_1"/>
<font color="Maroon" title="c12">b1</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="c10">CC</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:64_1"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="txt" href="#E12:64_1"><span class="lab"><font color="Green" title="E47">A6</font></span></a>, <a class="ref" href="zfmisc_1.html#T32" title="ZFMISC_1:th.32">ZFMISC_1:32</a></span>;<br/><a NAME="E2:64_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A16</font></span>: 
<font color="Maroon" title="c12">b1</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="c10">CC</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="#E17:64_1"><span class="lab"><font color="Green" title="E48">A7</font></span></a>, <a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</a></span>;<br/><span class="kw">assume </span><a NAME="E3:64_1_2_1"/><span class="lab"><font color="Green" title="E56">A17</font></span>: 
 not <font color="Maroon" title="c13">f</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="c10">CC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E4:64_1_2_1"/>
<font color="Maroon" title="c14">f1</font>,<font color="Maroon" title="c7">b</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="txt" href="#E5:64_1"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="txt" href="#E1:64_1_2"><span class="lab"><font color="Green" title="E53">A14</font></span></a>, <a class="ref" href="pencil_1.html#D3" target="_self" title="PENCIL_1:def.3">Def3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">lb</font> being   <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"> such that </span><br/><a NAME="E6:64_1_2_1"/><span class="lab"><font color="Green" title="E57">A18</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">f1</font>,<font color="Maroon" title="c7">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">lb</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_2_1"><span class="lab"><font color="Green" title="E56">A17</font></span></a>, <a class="txt" href="#E2:64_1_2_1"><span class="lab"><font color="Green" title="E55">A16</font></span></a>, <a class="ref" href="pencil_1.html#D1" target="_self" title="PENCIL_1:def.1">Def1</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c17">Lb</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="E8:64_1_2_1"/><span class="lab"><font color="Green" title="E58">A19</font></span>: 
<font color="Maroon" title="c16">lb</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="c17">Lb</font>
 <span class="kw">and </span><br/><a NAME="E9:64_1_2_1"/>
<font color="Maroon" title="c17">Lb</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="c17">Lb</font></span>)</span> is   <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="c17">Lb</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#T24" target="_self" title="PENCIL_1:th.24">Th24</a></span>;<br/><a NAME="E10:64_1_2_1"/><span class="lab"><font color="Green" title="E59">A20</font></span>: 
<font color="Maroon" title="c12">b1</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="c17">Lb</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:64_1_2_1"><span class="lab"><font color="Green" title="E57">A18</font></span></a>, <a class="txt" href="#E8:64_1_2_1"><span class="lab"><font color="Green" title="E58">A19</font></span></a>, <a class="ref" href="zfmisc_1.html#T32" title="ZFMISC_1:th.32">ZFMISC_1:32</a></span>;<br/><a NAME="E11:64_1_2_1"/><span class="lab"><font color="Green" title="E60">A21</font></span>: 
<font color="Maroon" title="c13">f</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="c17">Lb</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:64_1_2_1"><span class="lab"><font color="Green" title="E57">A18</font></span></a>, <a class="txt" href="#E8:64_1_2_1"><span class="lab"><font color="Green" title="E58">A19</font></span></a>, <a class="ref" href="zfmisc_1.html#T32" title="ZFMISC_1:th.32">ZFMISC_1:32</a></span>;<br/><a NAME="E12:64_1_2_1"/><span class="kw">then </span>
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c17">Lb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_2_1"><span class="lab"><font color="Green" title="E56">A17</font></span></a>, <a class="txt" href="#E2:64_1_2_1"><span class="lab"><font color="Green" title="E55">A16</font></span></a>, <a class="txt" href="#E10:64_1_2_1"><span class="lab"><font color="Green" title="E59">A20</font></span></a>, <a class="ref" href="pencil_1.html#T23" target="_self" title="PENCIL_1:th.23">Th23</a></span>;<br/><a NAME="E13:64_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">Lb</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="c10">CC</font></span>)</span> is 1 <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_2"><span class="lab"><font color="Green" title="E54">A15</font></span></a>, <a class="ref" href="pencil_1.html#T12" target="_self" title="PENCIL_1:th.12">Th12</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">cb</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E15:64_1_2_1"/><span class="lab"><font color="Green" title="E61">A22</font></span>: 
<font color="Maroon" title="c17">Lb</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="c10">CC</font></span>)</span> <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"><font color="Maroon" title="c18">cb</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T131" title="ZFMISC_1:th.131">ZFMISC_1:131</a></span>;<br/><a NAME="E16:64_1_2_1"/><span class="lab"><font color="Green" title="E62">A23</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c17">Lb</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="E17:64_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">b1</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="c10">CC</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">cb</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:64_1_2_1"><span class="lab"><font color="Green" title="E59">A20</font></span></a>, <a class="txt" href="#E15:64_1_2_1"><span class="lab"><font color="Green" title="E61">A22</font></span></a>, <a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</a></span>;<br/><a NAME="E18:64_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A24</font></span>: 
<font color="Maroon" title="c12">b1</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="c10">CC</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">cb</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E19:64_1_2_1"/>
<font color="Maroon" title="c11">a1</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="c10">CC</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:64_1"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="txt" href="#E12:64_1"><span class="lab"><font color="Green" title="E47">A6</font></span></a>, <a class="ref" href="zfmisc_1.html#T32" title="ZFMISC_1:th.32">ZFMISC_1:32</a></span>;<br/><a NAME="E20:64_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A25</font></span>: 
<font color="Maroon" title="c11">a1</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="c10">CC</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="#E17:64_1"><span class="lab"><font color="Green" title="E48">A7</font></span></a>, <a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</a></span>;<br/><a NAME="E21:64_1_2_1"/>
<font color="Maroon" title="c14">f1</font>,<font color="Maroon" title="c6">a</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="txt" href="#E4:64_1"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E1:64_1_2"><span class="lab"><font color="Green" title="E53">A14</font></span></a>, <a class="ref" href="pencil_1.html#D3" target="_self" title="PENCIL_1:def.3">Def3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">la</font> being   <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"> such that </span><br/><a NAME="E23:64_1_2_1"/><span class="lab"><font color="Green" title="E65">A26</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">f1</font>,<font color="Maroon" title="c6">a</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">la</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_2_1"><span class="lab"><font color="Green" title="E56">A17</font></span></a>, <a class="txt" href="#E20:64_1_2_1"><span class="lab"><font color="Green" title="E64">A25</font></span></a>, <a class="ref" href="pencil_1.html#D1" target="_self" title="PENCIL_1:def.1">Def1</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c20">La</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="E25:64_1_2_1"/><span class="lab"><font color="Green" title="E66">A27</font></span>: 
<font color="Maroon" title="c19">la</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="c20">La</font>
 <span class="kw">and </span><br/><a NAME="E26:64_1_2_1"/>
<font color="Maroon" title="c20">La</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="c20">La</font></span>)</span> is   <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="c20">La</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#T24" target="_self" title="PENCIL_1:th.24">Th24</a></span>;<br/><a NAME="E27:64_1_2_1"/><span class="lab"><font color="Green" title="E67">A28</font></span>: 
<font color="Maroon" title="c11">a1</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="c20">La</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:64_1_2_1"><span class="lab"><font color="Green" title="E65">A26</font></span></a>, <a class="txt" href="#E25:64_1_2_1"><span class="lab"><font color="Green" title="E66">A27</font></span></a>, <a class="ref" href="zfmisc_1.html#T32" title="ZFMISC_1:th.32">ZFMISC_1:32</a></span>;<br/><a NAME="E28:64_1_2_1"/><span class="lab"><font color="Green" title="E68">A29</font></span>: 
<font color="Maroon" title="c13">f</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="c20">La</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:64_1_2_1"><span class="lab"><font color="Green" title="E65">A26</font></span></a>, <a class="txt" href="#E25:64_1_2_1"><span class="lab"><font color="Green" title="E66">A27</font></span></a>, <a class="ref" href="zfmisc_1.html#T32" title="ZFMISC_1:th.32">ZFMISC_1:32</a></span>;<br/><a NAME="E29:64_1_2_1"/><span class="kw">then </span>
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c20">La</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_2_1"><span class="lab"><font color="Green" title="E56">A17</font></span></a>, <a class="txt" href="#E20:64_1_2_1"><span class="lab"><font color="Green" title="E64">A25</font></span></a>, <a class="txt" href="#E27:64_1_2_1"><span class="lab"><font color="Green" title="E67">A28</font></span></a>, <a class="ref" href="pencil_1.html#T23" target="_self" title="PENCIL_1:th.23">Th23</a></span>;<br/><a NAME="E30:64_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">La</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="c10">CC</font></span>)</span> is 1 <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_2"><span class="lab"><font color="Green" title="E54">A15</font></span></a>, <a class="ref" href="pencil_1.html#T12" target="_self" title="PENCIL_1:th.12">Th12</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">ca</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E32:64_1_2_1"/><span class="lab"><font color="Green" title="E69">A30</font></span>: 
<font color="Maroon" title="c20">La</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="c10">CC</font></span>)</span> <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"><font color="Maroon" title="c21">ca</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T131" title="ZFMISC_1:th.131">ZFMISC_1:131</a></span>;<br/><a NAME="E33:64_1_2_1"/><span class="lab"><font color="Green" title="E70">A31</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">La</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="E34:64_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">a1</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="c10">CC</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c21">ca</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:64_1_2_1"><span class="lab"><font color="Green" title="E67">A28</font></span></a>, <a class="txt" href="#E32:64_1_2_1"><span class="lab"><font color="Green" title="E69">A30</font></span></a>, <a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</a></span>;<br/><a NAME="E35:64_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A32</font></span>: 
<font color="Maroon" title="c21">ca</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">cb</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:64_1"><span class="lab"><font color="Green" title="E51">A10</font></span></a>, <a class="txt" href="#E18:64_1_2_1"><span class="lab"><font color="Green" title="E63">A24</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E36:64_1_2_1"/>
<font color="Maroon" title="c13">f</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="c10">CC</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c21">ca</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:64_1_2_1"><span class="lab"><font color="Green" title="E68">A29</font></span></a>, <a class="txt" href="#E32:64_1_2_1"><span class="lab"><font color="Green" title="E69">A30</font></span></a>, <a class="txt" href="#E33:64_1_2_1"><span class="lab"><font color="Green" title="E70">A31</font></span></a>, <a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</a></span>;<br/><a NAME="E37:64_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A33</font></span>: 
<font color="Maroon" title="c13">f</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="c10">CC</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">cb</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:64_1_2_1"><span class="lab"><font color="Green" title="E71">A32</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E38:64_1_2_1"/>
<font color="Maroon" title="c13">f</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="c10">CC</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">cb</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:64_1_2_1"><span class="lab"><font color="Green" title="E60">A21</font></span></a>, <a class="txt" href="#E15:64_1_2_1"><span class="lab"><font color="Green" title="E61">A22</font></span></a>, <a class="txt" href="#E16:64_1_2_1"><span class="lab"><font color="Green" title="E62">A23</font></span></a>, <a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</a></span>;<br/><span class="kw">hence </span><a NAME="E39:64_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:64_1_2_1"><span class="lab"><font color="Green" title="E72">A33</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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">hence </span><a NAME="E5:64_1_2"/>
<font color="Maroon" title="c13">f</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="c10">CC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
 ; <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="E23:64_1"/>
( <font color="Maroon" title="c11">a1</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="c10">CC</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> (<font color="Maroon" title="c3">S</font>,<span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c10">CC</font></span>)</span>) &amp; <font color="Maroon" title="c12">b1</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="c10">CC</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> (<font color="Maroon" title="c3">S</font>,<span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c10">CC</font></span>)</span>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E5:64_1"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="ref" href="card_3.html#D6" title="CARD_3:def.6">CARD_3:def 6</a></span>;<br/>
<a NAME="E24:64_1"/><span class="kw">then </span>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="card_3.html#K5" title="CARD_3:func.5">pi</a> (<font color="Maroon" title="c3">S</font>,<span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c10">CC</font></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:64_1"><span class="lab"><font color="Green" title="E51">A10</font></span></a>, <a class="ref" href="pencil_1.html#T2" target="_self" title="PENCIL_1:th.2">Th2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">p</font> =  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> (<font color="Maroon" title="c3">S</font>,<span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c10">CC</font></span>)</span>) as   non  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#T4" target="_self" title="PENCIL_1:th.4">Th4</a></span>;<br/>
<a NAME="E26:64_1"/>
<font color="Maroon" title="c10">CC</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="c10">CC</font></span>)</span>,<font color="Maroon" title="c13">p</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">i</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="pboole.html#D2" title="PBOOLE:def.2">PBOOLE:def 2</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="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c10">CC</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c10">CC</font></span>)</span>,<font color="Maroon" title="c13">p</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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="c14">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_1_3"/><span class="lab"><font color="Green" title="E53">A34</font></span>: 
<font color="Maroon" title="c14">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"> <span class="p1">(<span class="default"><font color="Maroon" title="c10">CC</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c10">CC</font></span>)</span>,<font color="Maroon" title="c13">p</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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="c14">i</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_3_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="E1:64_1_3_1"/>
( <font color="Maroon" title="c14">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="c10">CC</font> or <font color="Maroon" title="c14">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="c10">CC</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:64_1_3_1_1"/><span class="lab"><font color="Green" title="E54">A35</font></span>: 
<font color="Maroon" title="c14">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="c10">CC</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c10">CC</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c10">CC</font></span>)</span>,<font color="Maroon" title="c13">p</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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="c14">i</font></span><br/><div class="add"><a NAME="E2:64_1_3_1_1"/><span class="lab"><font color="Green" title="E55">A36</font></span>: 
<font color="Maroon" title="c10">CC</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="E3:64_1_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">CC</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c10">CC</font></span>)</span>,<font color="Maroon" title="c13">p</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">CC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_3_1_1"><span class="lab"><font color="Green" title="E54">A35</font></span></a>, <a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></span>;<br/><span class="kw">hence </span><a NAME="E4:64_1_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">CC</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c10">CC</font></span>)</span>,<font color="Maroon" title="c13">p</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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="c14">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_3"><span class="lab"><font color="Green" title="E53">A34</font></span></a>, <a class="txt" href="#E2:64_1_3_1_1"><span class="lab"><font color="Green" title="E55">A36</font></span></a>, <a class="ref" href="pboole.html#D2" title="PBOOLE:def.2">PBOOLE:def 2</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="64_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:64_1_3_1_2"/><span class="lab"><font color="Green" title="E54">A37</font></span>: 
<font color="Maroon" title="c14">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="c10">CC</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c10">CC</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c10">CC</font></span>)</span>,<font color="Maroon" title="c13">p</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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="c14">i</font></span><br/><div class="add"><a NAME="E2:64_1_3_1_2"/><span class="lab"><font color="Green" title="E55">A38</font></span>: 
<font color="Maroon" title="c13">p</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="c14">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_3_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">y</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="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">p</font> or <font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c14">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_1_3_1_2_1"/>
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">p</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="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c14">i</font></span><br/>

<a NAME="E2:64_1_3_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A39</font></span>: 
 ex <font color="Olive" title="b1">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font> &amp; <font color="Maroon" title="c15">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_3_1_2"><span class="lab"><font color="Green" title="E54">A37</font></span></a>, <a class="ref" href="card_3.html#D6" title="CARD_3:def.6">CARD_3:def 6</a></span>;<br/>
<a NAME="E3:64_1_3_1_2_1"/>
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_3"><span class="lab"><font color="Green" title="E53">A34</font></span></a>, <a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:64_1_3_1_2_1"/>
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c14">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_1_3_1_2_1"><span class="lab"><font color="Green" title="E56">A39</font></span></a>, <a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</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="E3:64_1_3_1_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">CC</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">hence </span><a NAME="E4:64_1_3_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">CC</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c10">CC</font></span>)</span>,<font color="Maroon" title="c13">p</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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="c14">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_3_1_2"><span class="lab"><font color="Green" title="E54">A37</font></span></a>, <a class="txt" href="#E2:64_1_3_1_2"><span class="lab"><font color="Green" title="E55">A38</font></span></a>, <a class="ref" href="funct_7.html#T31" title="FUNCT_7:th.31">FUNCT_7:31</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><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">B</font> = <font color="Maroon" title="c10">CC</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="c10">CC</font></span>)</span>,<font color="Maroon" title="c13">p</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#T27" target="_self" title="PENCIL_1:th.27">Th27</a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c14">B</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="c3">S</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="c14">B</font> &amp; (  for <font color="Olive" title="b1">C</font> being   <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="c14">B</font></span>)</span></span>)</span>  st <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">B</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="c14">B</font></span>)</span> holds <br/>( <font color="Olive" title="b1">C</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Olive" title="b1">C</font> is  <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a>  ) ) )</span><br/>

<a NAME="E28:64_1"/><span class="lab"><font color="Green" title="E53">A40</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">B</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="E29:64_1"/><span class="lab"><font color="Green" title="E54">A41</font></span>: 
<font color="Maroon" title="c14">B</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="c10">CC</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:64_1"><span class="lab"><font color="Green" title="E48">A7</font></span></a>, <a class="ref" href="funct_7.html#T31" title="FUNCT_7:th.31">FUNCT_7:31</a></span>;<br/>
<span class="kw">thus </span><a NAME="E30:64_1"/><span class="lab"><font color="Green" title="E55">A42</font></span>: 
<font color="Maroon" title="c3">S</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="c14">B</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">C</font> being   <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="c14">B</font></span>)</span></span>)</span>  st <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">B</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="c14">B</font></span>)</span> holds <br/>( <font color="Olive" title="b1">C</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Olive" title="b1">C</font> is  <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:64_1_4"/>
<font color="Maroon" title="c3">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c14">B</font>
  <span class="comment"><font color="firebrick">:: according to </font></span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c14">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">S</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">e</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="c15">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font> or <font color="Maroon" title="c15">e</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="c14">B</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_1_4_1"/><span class="lab"><font color="Green" title="E56">A43</font></span>: 
<font color="Maroon" title="c15">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</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="c15">e</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="c14">B</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">f</font> = <font color="Maroon" title="c15">e</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="ref" href="pencil_1.html#T14" target="_self" title="PENCIL_1:th.14">Th14</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E57">A44</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_4_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">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="c17">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="c16">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:64_1_4_1_1"/>
<font color="Maroon" title="c17">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="c16">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">j</font> = <font color="Maroon" title="c17">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_4_1_1_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="E1:64_1_4_1_1_1"/>
( <font color="Maroon" title="c18">j</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="c10">CC</font> or <font color="Maroon" title="c18">j</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="c10">CC</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_4_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:64_1_4_1_1_1_1"/>
<font color="Maroon" title="c18">j</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="c10">CC</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="c16">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:64_1_4_1_1_1_1"/>
<font color="Maroon" title="c16">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:64_1"><span class="lab"><font color="Green" title="E54">A41</font></span></a>, <a class="txt" href="#E1:64_1_4_1"><span class="lab"><font color="Green" title="E56">A43</font></span></a>, <a class="ref" href="card_3.html#D6" title="CARD_3:def.6">CARD_3:def 6</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="64_1_4_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:64_1_4_1_1_1_2"/><span class="lab"><font color="Green" title="E58">A45</font></span>: 
<font color="Maroon" title="c18">j</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="c10">CC</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="c16">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:64_1_4_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">CC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:64_1"><span class="lab"><font color="Green" title="E52">A13</font></span></a>, <a class="txt" href="#E1:64_1_4_1"><span class="lab"><font color="Green" title="E56">A43</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:64_1_4_1_1_1_2"/>
<font color="Maroon" title="c16">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_4_1_1_1_2"><span class="lab"><font color="Green" title="E58">A45</font></span></a>, <a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</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="E4:64_1_4_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c16">f</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">hence </span><a NAME="E5:64_1_4_1"/>
<font color="Maroon" title="c15">e</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="c14">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:64_1"><span class="lab"><font color="Green" title="E53">A40</font></span></a>, <a class="txt" href="#E3:64_1_4_1"><span class="lab"><font color="Green" title="E57">A44</font></span></a>, <a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</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">let </span><font color="Maroon" title="c15">e</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="c15">e</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="c14">B</font> or <font color="Maroon" title="c15">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:64_1_4"/>
<font color="Maroon" title="c15">e</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="c14">B</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="c15">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E4:64_1_4"/><span class="lab"><font color="Green" title="E56">A46</font></span>: 
( <font color="Maroon" title="c15">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">f</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c16">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> )
 <span class="kw">and </span><br/><a NAME="E5:64_1_4"/><span class="lab"><font color="Green" title="E57">A47</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="c16">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:64_1"><span class="lab"><font color="Green" title="E53">A40</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="E6:64_1_4"/>
<font color="Maroon" title="c16">f</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="c10">CC</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:64_1"><span class="lab"><font color="Green" title="E54">A41</font></span></a>, <a class="txt" href="#E5:64_1_4"><span class="lab"><font color="Green" title="E57">A47</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E8:64_1_4"/><span class="lab"><font color="Green" title="E58">A48</font></span>: 
<font color="Maroon" title="c17">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font>
 <span class="kw">and </span><br/><a NAME="E9:64_1_4"/><span class="lab"><font color="Green" title="E59">A49</font></span>: 
<font color="Maroon" title="c16">f</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="c10">CC</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g</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="c10">CC</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D6" title="CARD_3:def.6">CARD_3:def 6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">g</font> = <font color="Maroon" title="c17">g</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="#E8:64_1_4"><span class="lab"><font color="Green" title="E58">A48</font></span></a></span>;<br/>
<div><span class="lab"><font color="Green" title="E60">A50</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_4_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c19">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="c19">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="c16">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:64_1_4_2"/>
<font color="Maroon" title="c19">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="c16">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">j</font> = <font color="Maroon" title="c19">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_4_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="E1:64_1_4_2_1"/>
( <font color="Maroon" title="c19">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="c10">CC</font> or <font color="Maroon" title="c19">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="c10">CC</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_4_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:64_1_4_2_1_1"/>
<font color="Maroon" title="c19">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="c10">CC</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="c16">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:64_1_4_2_1_1"/>
<font color="Maroon" title="c16">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:64_1_4"><span class="lab"><font color="Green" title="E59">A49</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_4_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:64_1_4_2_1_2"/><span class="lab"><font color="Green" title="E61">A51</font></span>: 
<font color="Maroon" title="c19">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="c10">CC</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="c16">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:64_1_4_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">CC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font> is 1 <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#T12" target="_self" title="PENCIL_1:th.12">Th12</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:64_1_4_2_1_2"/><span class="lab"><font color="Green" title="E62">A52</font></span>: 
<font color="Maroon" title="c10">CC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">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"><font color="Maroon" title="c21">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T131" title="ZFMISC_1:th.131">ZFMISC_1:131</a></span>;<br/><a NAME="E5:64_1_4_2_1_2"/><span class="lab"><font color="Green" title="E63">A53</font></span>: 
<font color="Maroon" title="c18">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">CC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:64_1"><span class="lab"><font color="Green" title="E52">A13</font></span></a>, <a class="txt" href="#E8:64_1_4"><span class="lab"><font color="Green" title="E58">A48</font></span></a>, <a class="txt" href="#E1:64_1_4_2_1_2"><span class="lab"><font color="Green" title="E61">A51</font></span></a></span>;<br/><a NAME="E6:64_1_4_2_1_2"/>
<font color="Maroon" title="c16">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:64_1_4"><span class="lab"><font color="Green" title="E57">A47</font></span></a></span>;<br/><a NAME="E7:64_1_4_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c21">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_4_2_1_2"><span class="lab"><font color="Green" title="E61">A51</font></span></a>, <a class="txt" href="#E4:64_1_4_2_1_2"><span class="lab"><font color="Green" title="E62">A52</font></span></a>, <a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c16">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> = 
<font color="Maroon" title="c21">c</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c18">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1_4_2_1_2"><span class="lab"><font color="Green" title="E62">A52</font></span></a>, <a class="txt" href="#E5:64_1_4_2_1_2"><span class="lab"><font color="Green" title="E63">A53</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>
;<br/> <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="E12:64_1_4"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">g</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">hence </span><a NAME="E13:64_1_4"/>
<font color="Maroon" title="c15">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1_4"><span class="lab"><font color="Green" title="E56">A46</font></span></a>, <a class="txt" href="#E8:64_1_4"><span class="lab"><font color="Green" title="E58">A48</font></span></a>, <a class="txt" href="#E11:64_1_4"><span class="lab"><font color="Green" title="E60">A50</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</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">let </span><font color="Maroon" title="c15">SS</font> be   <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="c14">B</font></span>)</span></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="c15">SS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">B</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="c14">B</font></span>)</span> implies ( <font color="Maroon" title="c15">SS</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Maroon" title="c15">SS</font> is  <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E31:64_1"/><span class="lab"><font color="Green" title="E56">A54</font></span>: 
<font color="Maroon" title="c15">SS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">B</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="c14">B</font></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="c15">SS</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Maroon" title="c15">SS</font> is  <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E32:64_1"/>
<font color="Maroon" title="c15">SS</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</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">SS</font> is  <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">q</font>, <font color="Maroon" title="c17">r</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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="c14">B</font></span>)</span></span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="pencil_1.html#D3" target="_self" title="PENCIL_1:def.3">PENCIL_1:def 3</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="c16">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">SS</font> &amp; <font color="Maroon" title="c17">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">SS</font> implies <font color="Maroon" title="c16">q</font>,<font color="Maroon" title="c17">r</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:64_1_5"/><span class="lab"><font color="Green" title="E57">A55</font></span>: 
<font color="Maroon" title="c16">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">SS</font>
 <span class="kw">and </span><br/><a NAME="E2:64_1_5"/><span class="lab"><font color="Green" title="E58">A56</font></span>: 
<font color="Maroon" title="c17">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">SS</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="c16">q</font>,<font color="Maroon" title="c17">r</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a> </span><br/>

<span class="kw">thus </span><a NAME="E3:64_1_5"/>
<font color="Maroon" title="c16">q</font>,<font color="Maroon" title="c17">r</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_5_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_5_1_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="E1:64_1_5_1_1"/>
( <font color="Maroon" title="c16">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">r</font> or <font color="Maroon" title="c16">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">r</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_5_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:64_1_5_1_1_1"/>
<font color="Maroon" title="c16">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">r</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="c16">q</font>,<font color="Maroon" title="c17">r</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:64_1_5_1_1_1"/>
<font color="Maroon" title="c16">q</font>,<font color="Maroon" title="c17">r</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#D1" target="_self" title="PENCIL_1:def.1">Def1</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="64_1_5_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:64_1_5_1_1_2"/><span class="lab"><font color="Green" title="E59">A57</font></span>: 
<font color="Maroon" title="c16">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">r</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="c16">q</font>,<font color="Maroon" title="c17">r</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a> </span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c18">R</font> = <font color="Maroon" title="c11">a1</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="c14">B</font></span>)</span>,<font color="Maroon" title="c17">r</font>) as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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="ref" href="pencil_1.html#T25" target="_self" title="PENCIL_1:th.25">Th25</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">Q</font> = <font color="Maroon" title="c11">a1</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="c14">B</font></span>)</span>,<font color="Maroon" title="c16">q</font>) as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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="ref" href="pencil_1.html#T25" target="_self" title="PENCIL_1:th.25">Th25</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">Q1</font> = <font color="Maroon" title="c19">Q</font>, <font color="Maroon" title="c21">R1</font> = <font color="Maroon" title="c18">R</font> as   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> ;<br/><div><span class="lab"><font color="Green" title="E60">A58</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_5_1_1_2_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:64_1_5_1_1_2_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">a1</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="E2:64_1_5_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A59</font></span>: 
<font color="Maroon" title="c20">Q1</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="c14">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">q</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="E3:64_1_5_1_1_2_1"/><span class="lab"><font color="Green" title="E62">A60</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">a1</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">assume </span><a NAME="E4:64_1_5_1_1_2_1"/>
<font color="Maroon" title="c19">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">hence </span><a NAME="E5:64_1_5_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_5_1_1_2"><span class="lab"><font color="Green" title="E59">A57</font></span></a>, <a class="txt" href="#E2:64_1_5_1_1_2_1"><span class="lab"><font color="Green" title="E61">A59</font></span></a>, <a class="txt" href="#E3:64_1_5_1_1_2_1"><span class="lab"><font color="Green" title="E62">A60</font></span></a>, <a class="ref" href="funct_7.html#T31" title="FUNCT_7:th.31">FUNCT_7:31</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><a NAME="E6:64_1_5_1_1_2"/><span class="lab"><font color="Green" title="E61">A61</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>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">B</font> holds <br/><font color="Maroon" title="c20">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</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="64_1_5_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c22">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="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">B</font> implies <font color="Maroon" title="c20">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_1_5_1_1_2_2"/><span class="lab"><font color="Green" title="E62">A62</font></span>: 
<font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">B</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="c20">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span><br/>

<a NAME="E2:64_1_5_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="E3:64_1_5_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A63</font></span>: 
<font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">a1</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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_5_1_1_2_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="E1:64_1_5_1_1_2_2_1"/>
( <font color="Maroon" title="c22">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="c14">B</font> or <font color="Maroon" title="c22">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="c14">B</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_5_1_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:64_1_5_1_1_2_2_1_1"/>
<font color="Maroon" title="c22">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="c14">B</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="c20">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span><br/><div class="add"><a NAME="E2:64_1_5_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">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/><span class="kw">hence </span><a NAME="E3:64_1_5_1_1_2_2_1_1"/>
<font color="Maroon" title="c20">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E30:64_1"><span class="lab"><font color="Green" title="E55">A42</font></span></a>, <a class="txt" href="#E1:64_1_5_1_1_2_2"><span class="lab"><font color="Green" title="E62">A62</font></span></a>, <a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</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="64_1_5_1_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:64_1_5_1_1_2_2_1_2"/>
<font color="Maroon" title="c22">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="c14">B</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="c20">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:64_1_5_1_1_2_2_1_2"/>
<font color="Maroon" title="c20">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:64_1"><span class="lab"><font color="Green" title="E56">A54</font></span></a>, <a class="txt" href="#E1:64_1_5"><span class="lab"><font color="Green" title="E57">A55</font></span></a>, <a class="txt" href="#E3:64_1_5_1_1_2_2"><span class="lab"><font color="Green" title="E63">A63</font></span></a>, <a class="ref" href="funct_7.html#T31" title="FUNCT_7:th.31">FUNCT_7:31</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><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E7:64_1_5_1_1_2"/><span class="lab"><font color="Green" title="E62">A64</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>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">B</font> holds <br/><font color="Maroon" title="c21">R1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</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="64_1_5_1_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c22">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="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">B</font> implies <font color="Maroon" title="c21">R1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_1_5_1_1_2_3"/><span class="lab"><font color="Green" title="E63">A65</font></span>: 
<font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">B</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="c21">R1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span><br/>

<a NAME="E2:64_1_5_1_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="E3:64_1_5_1_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A66</font></span>: 
<font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">a1</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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_5_1_1_2_3_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="E1:64_1_5_1_1_2_3_1"/>
( <font color="Maroon" title="c22">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="c14">B</font> or <font color="Maroon" title="c22">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="c14">B</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_5_1_1_2_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:64_1_5_1_1_2_3_1_1"/>
<font color="Maroon" title="c22">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="c14">B</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="c21">R1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span><br/><div class="add"><a NAME="E2:64_1_5_1_1_2_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">R1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">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/><span class="kw">hence </span><a NAME="E3:64_1_5_1_1_2_3_1_1"/>
<font color="Maroon" title="c21">R1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E30:64_1"><span class="lab"><font color="Green" title="E55">A42</font></span></a>, <a class="txt" href="#E1:64_1_5_1_1_2_3"><span class="lab"><font color="Green" title="E63">A65</font></span></a>, <a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</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="64_1_5_1_1_2_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:64_1_5_1_1_2_3_1_2"/>
<font color="Maroon" title="c22">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="c14">B</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="c21">R1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:64_1_5_1_1_2_3_1_2"/>
<font color="Maroon" title="c21">R1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:64_1"><span class="lab"><font color="Green" title="E56">A54</font></span></a>, <a class="txt" href="#E2:64_1_5"><span class="lab"><font color="Green" title="E58">A56</font></span></a>, <a class="txt" href="#E3:64_1_5_1_1_2_3"><span class="lab"><font color="Green" title="E64">A66</font></span></a>, <a class="ref" href="funct_7.html#T31" title="FUNCT_7:th.31">FUNCT_7:31</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><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div> <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c21">R1</font> = 
<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 href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">B</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="E9:64_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A67</font></span>: 
<font color="Maroon" title="c18">R</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="c14">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:64_1_5_1_1_2"><span class="lab"><font color="Green" title="E62">A64</font></span></a>, <a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</a></span>;<br/> <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">Q1</font> = 
<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 href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">B</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="E11:64_1_5_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c19">Q</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="c14">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:64_1_5_1_1_2"><span class="lab"><font color="Green" title="E61">A61</font></span></a>, <a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</a></span>;<br/><a NAME="E12:64_1_5_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c19">Q</font>,<font color="Maroon" title="c18">R</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="txt" href="#E30:64_1"><span class="lab"><font color="Green" title="E55">A42</font></span></a>, <a class="txt" href="#E9:64_1_5_1_1_2"><span class="lab"><font color="Green" title="E63">A67</font></span></a>, <a class="ref" href="pencil_1.html#D3" target="_self" title="PENCIL_1:def.3">Def3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">L</font> being   <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"> such that </span><br/><a NAME="E14:64_1_5_1_1_2"/><span class="lab"><font color="Green" title="E64">A68</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c19">Q</font>,<font color="Maroon" title="c18">R</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c22">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:64_1_5_1_1_2"><span class="lab"><font color="Green" title="E60">A58</font></span></a>, <a class="ref" href="pencil_1.html#D1" target="_self" title="PENCIL_1:def.1">Def1</a></span>;<br/><a NAME="E15:64_1_5_1_1_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">a1</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="E16:64_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A69</font></span>: 
<font color="Maroon" title="c21">R1</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="c14">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">r</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/><span class="kw">consider </span><font color="Maroon" title="c23">L1</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="E18:64_1_5_1_1_2"/><span class="lab"><font color="Green" title="E66">A70</font></span>: 
<font color="Maroon" title="c22">L</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="c23">L1</font>
 <span class="kw">and </span><br/><a NAME="E19:64_1_5_1_1_2"/><span class="lab"><font color="Green" title="E67">A71</font></span>: 
<font color="Maroon" title="c23">L1</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="c23">L1</font></span>)</span> is   <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="c23">L1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#T24" target="_self" title="PENCIL_1:th.24">Th24</a></span>;<br/><a NAME="E20:64_1_5_1_1_2"/><span class="lab"><font color="Green" title="E68">A72</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c23">L1</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="E21:64_1_5_1_1_2"/>
<font color="Maroon" title="c20">Q1</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="c23">L1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:64_1_5_1_1_2"><span class="lab"><font color="Green" title="E64">A68</font></span></a>, <a class="txt" href="#E18:64_1_5_1_1_2"><span class="lab"><font color="Green" title="E66">A70</font></span></a>, <a class="ref" href="zfmisc_1.html#T32" title="ZFMISC_1:th.32">ZFMISC_1:32</a></span>;<br/><a NAME="E22:64_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A73</font></span>: 
<font color="Maroon" title="c20">Q1</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="c14">B</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">L1</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="c14">B</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:64_1_5_1_1_2"><span class="lab"><font color="Green" title="E68">A72</font></span></a>, <a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</a></span>;<br/><a NAME="E23:64_1_5_1_1_2"/><span class="lab"><font color="Green" title="E70">A74</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c23">L1</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="E24:64_1_5_1_1_2"/>
<font color="Maroon" title="c21">R1</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="c23">L1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:64_1_5_1_1_2"><span class="lab"><font color="Green" title="E64">A68</font></span></a>, <a class="txt" href="#E18:64_1_5_1_1_2"><span class="lab"><font color="Green" title="E66">A70</font></span></a>, <a class="ref" href="zfmisc_1.html#T32" title="ZFMISC_1:th.32">ZFMISC_1:32</a></span>;<br/><a NAME="E25:64_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A75</font></span>: 
<font color="Maroon" title="c21">R1</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="c14">B</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">L1</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="c14">B</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:64_1_5_1_1_2"><span class="lab"><font color="Green" title="E70">A74</font></span></a>, <a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_5_1_1_2_6"><span class="kw">now </span></a><div class="add"><a NAME="E1:64_1_5_1_1_2_6"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">a1</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="E2:64_1_5_1_1_2_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A76</font></span>: 
<font color="Maroon" title="c20">Q1</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="c14">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">q</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="E3:64_1_5_1_1_2_6"/><span class="lab"><font color="Green" title="E73">A77</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">a1</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">assume </span><a NAME="E4:64_1_5_1_1_2_6"/>
<font color="Maroon" title="c20">Q1</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="c14">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">R1</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="c14">B</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">hence </span><a NAME="E5:64_1_5_1_1_2_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_5_1_1_2"><span class="lab"><font color="Green" title="E59">A57</font></span></a>, <a class="txt" href="#E2:64_1_5_1_1_2_6"><span class="lab"><font color="Green" title="E72">A76</font></span></a>, <a class="txt" href="#E3:64_1_5_1_1_2_6"><span class="lab"><font color="Green" title="E73">A77</font></span></a>, <a class="ref" href="funct_7.html#T31" title="FUNCT_7:th.31">FUNCT_7:31</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><a NAME="E27:64_1_5_1_1_2"/><span class="kw">then </span>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c14">B</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:64_1_5_1_1_2"><span class="lab"><font color="Green" title="E69">A73</font></span></a>, <a class="txt" href="#E25:64_1_5_1_1_2"><span class="lab"><font color="Green" title="E71">A75</font></span></a>, <a class="ref" href="pencil_1.html#T2" target="_self" title="PENCIL_1:th.2">Th2</a></span>;<br/><a NAME="E28:64_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A78</font></span>: 
 not <font color="Maroon" title="c23">L1</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="c14">B</font></span>)</span> 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" target="_self" title="PENCIL_1:th.4">Th4</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c24">LI</font> = <font color="Maroon" title="c23">L1</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="c23">L1</font></span>)</span> as   <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="c14">B</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:64_1_5_1_1_2"><span class="lab"><font color="Green" title="E67">A71</font></span></a>, <a class="ref" href="pencil_1.html#D21" target="_self" title="PENCIL_1:def.21">Def21</a></span>;<br/><a NAME="E30:64_1_5_1_1_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">a1</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="E31:64_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A79</font></span>: 
<font color="Maroon" title="c20">Q1</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="c14">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">q</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="E32:64_1_5_1_1_2"/>
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c14">B</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="c23">L1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:64_1_5_1_1_2"><span class="lab"><font color="Green" title="E72">A78</font></span></a>, <a class="ref" href="pencil_1.html#D21" target="_self" title="PENCIL_1:def.21">Def21</a></span>;<br/><a NAME="E33:64_1_5_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c16">q</font>,<font color="Maroon" title="c17">r</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c24">LI</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:64_1_5_1_1_2"><span class="lab"><font color="Green" title="E69">A73</font></span></a>, <a class="txt" href="#E25:64_1_5_1_1_2"><span class="lab"><font color="Green" title="E71">A75</font></span></a>, <a class="txt" href="#E31:64_1_5_1_1_2"><span class="lab"><font color="Green" title="E73">A79</font></span></a>, <a class="txt" href="#E16:64_1_5_1_1_2"><span class="lab"><font color="Green" title="E65">A69</font></span></a>, <a class="ref" href="zfmisc_1.html#T32" title="ZFMISC_1:th.32">ZFMISC_1:32</a></span>;<br/><span class="kw">hence </span><a NAME="E34:64_1_5_1_1_2"/>
<font color="Maroon" title="c16">q</font>,<font color="Maroon" title="c17">r</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#D1" target="_self" title="PENCIL_1:def.1">Def1</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>


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

<span class="kw">thus </span><a NAME="E33:64_1"/>
<font color="Maroon" title="c15">SS</font> is  <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">L</font> be   <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="c14">B</font></span>)</span></span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="pencil_1.html#D2" target="_self" title="PENCIL_1:def.2">PENCIL_1:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">L</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c15">SS</font></span>)</span> implies <font color="Maroon" title="c16">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">SS</font> )</span><br/>

<a NAME="E1:64_1_6"/><span class="lab"><font color="Green" title="E57">A80</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">B</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="E2:64_1_6"/>
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="c16">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#D6" target="_self" title="PENCIL_1:def.6">Def6</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">L1</font> = <font color="Maroon" title="c16">L</font> as   non  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#T4" target="_self" title="PENCIL_1:th.4">Th4</a></span>;<br/>
<a NAME="E4:64_1_6"/>
<font color="Maroon" title="c16">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <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="c14">B</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E5:64_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A81</font></span>: 
<font color="Maroon" title="c16">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c14">B</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E6:64_1_6"/>
<font color="Maroon" title="c14">B</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="c14">B</font></span>)</span>,<font color="Maroon" title="c17">L1</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_6_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">o</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="pboole.html#D2" title="PBOOLE:def.2">PBOOLE:def 2</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="c18">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c17">L1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o</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="c18">o</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_1_6_1"/><span class="lab"><font color="Green" title="E59">A82</font></span>: 
<font color="Maroon" title="c18">o</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c17">L1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o</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="c18">o</font></span><br/>

<span class="kw">thus </span><a NAME="E2:64_1_6_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c17">L1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o</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="c18">o</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_6_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_6_1_1_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="E1:64_1_6_1_1_1"/>
( <font color="Maroon" title="c18">o</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="c14">B</font> or <font color="Maroon" title="c18">o</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="c14">B</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_6_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:64_1_6_1_1_1_1"/><span class="lab"><font color="Green" title="E60">A83</font></span>: 
<font color="Maroon" title="c18">o</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="c14">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c17">L1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o</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="c18">o</font></span><br/><div class="add"><a NAME="E2:64_1_6_1_1_1_1"/><span class="lab"><font color="Green" title="E61">A84</font></span>: 
<font color="Maroon" title="c14">B</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="E3:64_1_6_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c17">L1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_6_1_1_1_1"><span class="lab"><font color="Green" title="E60">A83</font></span></a>, <a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></span>;<br/><span class="kw">hence </span><a NAME="E4:64_1_6_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c17">L1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o</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="c18">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_6_1"><span class="lab"><font color="Green" title="E59">A82</font></span></a>, <a class="txt" href="#E2:64_1_6_1_1_1_1"><span class="lab"><font color="Green" title="E61">A84</font></span></a>, <a class="ref" href="pboole.html#D2" title="PBOOLE:def.2">PBOOLE:def 2</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="64_1_6_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:64_1_6_1_1_1_2"/><span class="lab"><font color="Green" title="E60">A85</font></span>: 
<font color="Maroon" title="c18">o</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="c14">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c17">L1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o</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="c18">o</font></span><br/><div class="add"><a NAME="E2:64_1_6_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c17">L1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:64_1"><span class="lab"><font color="Green" title="E53">A40</font></span></a>, <a class="ref" href="funct_7.html#T31" title="FUNCT_7:th.31">FUNCT_7:31</a></span>;<br/><span class="kw">hence </span><a NAME="E3:64_1_6_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c17">L1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o</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="c18">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:64_1_6"><span class="lab"><font color="Green" title="E58">A81</font></span></a>, <a class="txt" href="#E1:64_1_6_1_1_1_2"><span class="lab"><font color="Green" title="E60">A85</font></span></a>, <a class="ref" href="yellow_6.html#T2" title="YELLOW_6:th.2">YELLOW_6:2</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>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">L2</font> = <font color="Maroon" title="c14">B</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="c14">B</font></span>)</span>,<font color="Maroon" title="c17">L1</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#T27" target="_self" title="PENCIL_1:th.27">Th27</a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></span>;<br/>
<a NAME="E8:64_1_6"/><span class="lab"><font color="Green" title="E59">A86</font></span>: 
<font color="Maroon" title="c18">L2</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="c14">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:64_1"><span class="lab"><font color="Green" title="E53">A40</font></span></a>, <a class="ref" href="funct_7.html#T31" title="FUNCT_7:th.31">FUNCT_7:31</a></span>;<br/>
<a NAME="E9:64_1_6"/><span class="kw">then </span>
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c14">B</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="c18">L2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#D21" target="_self" title="PENCIL_1:def.21">Def21</a></span>;<br/>
<a NAME="E10:64_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c18">L2</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="c18">L2</font></span>)</span> is   <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="c18">L2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_6"><span class="lab"><font color="Green" title="E57">A80</font></span></a>, <a class="ref" href="funct_7.html#T31" title="FUNCT_7:th.31">FUNCT_7:31</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">L3</font> =  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c18">L2</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="ref" href="pencil_1.html#T24" target="_self" title="PENCIL_1:th.24">Th24</a></span>;<br/>
<span class="kw">assume </span><a NAME="E12:64_1_6"/>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">L</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c15">SS</font></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="c16">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">SS</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">x</font>, <font color="Maroon" title="c21">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E14:64_1_6"/><span class="lab"><font color="Green" title="E60">A87</font></span>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">L</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c15">SS</font>
 <span class="kw">and </span><br/><a NAME="E15:64_1_6"/><span class="lab"><font color="Green" title="E61">A88</font></span>: 
<font color="Maroon" title="c21">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">L</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c15">SS</font>
 <span class="kw">and </span><br/><a NAME="E16:64_1_6"/><span class="lab"><font color="Green" title="E62">A89</font></span>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c21">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#T2" target="_self" title="PENCIL_1:th.2">Th2</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c22">y1</font> = <font color="Maroon" title="c11">a1</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="c14">B</font></span>)</span>,<font color="Maroon" title="c21">y</font>);<br/>
<a NAME="E17:64_1_6"/><span class="lab"><font color="Green" title="E63">A90</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c21">y</font>)</span>)</span> <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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_6_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c23">o</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="c23">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c21">y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:64_1_6_2"/><span class="lab"><font color="Green" title="E64">A91</font></span>: 
<font color="Maroon" title="c23">o</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c21">y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_6_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="E1:64_1_6_2_1"/>
( <font color="Maroon" title="c23">o</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="c14">B</font> or <font color="Maroon" title="c23">o</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="c14">B</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_6_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:64_1_6_2_1_1"/>
<font color="Maroon" title="c23">o</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="c14">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c21">y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:64_1_6_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c21">y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">o</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/><span class="kw">hence </span><a NAME="E3:64_1_6_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c21">y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E28:64_1"><span class="lab"><font color="Green" title="E53">A40</font></span></a>, <a class="txt" href="#E30:64_1"><span class="lab"><font color="Green" title="E55">A42</font></span></a>, <a class="txt" href="#E1:64_1_6_2"><span class="lab"><font color="Green" title="E64">A91</font></span></a>, <a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</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="64_1_6_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:64_1_6_2_1_2"/><span class="lab"><font color="Green" title="E65">A92</font></span>: 
<font color="Maroon" title="c23">o</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="c14">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c21">y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:64_1_6_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c21">y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:64_1"><span class="lab"><font color="Green" title="E49">A8</font></span></a>, <a class="ref" href="funct_7.html#T31" title="FUNCT_7:th.31">FUNCT_7:31</a></span>;<br/><span class="kw">hence </span><a NAME="E3:64_1_6_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c21">y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:64_1"><span class="lab"><font color="Green" title="E56">A54</font></span></a>, <a class="txt" href="#E15:64_1_6"><span class="lab"><font color="Green" title="E61">A88</font></span></a>, <a class="txt" href="#E1:64_1_6_2_1_2"><span class="lab"><font color="Green" title="E65">A92</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0: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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E19:64_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A93</font></span>: 
<font color="Maroon" title="c11">a1</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="c14">B</font></span>)</span>,<font color="Maroon" title="c21">y</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:64_1"><span class="lab"><font color="Green" title="E53">A40</font></span></a>, <a class="txt" href="#E30:64_1"><span class="lab"><font color="Green" title="E55">A42</font></span></a>, <a class="txt" href="#E17:64_1_6"><span class="lab"><font color="Green" title="E63">A90</font></span></a>, <a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</a></span>;<br/>
<a NAME="E20:64_1_6"/><span class="lab"><font color="Green" title="E65">A94</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">L2</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_6_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c23">o</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="c23">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">L2</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c21">y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:64_1_6_3"/><span class="lab"><font color="Green" title="E66">A95</font></span>: 
<font color="Maroon" title="c23">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">L2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c21">y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_6_3_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="E1:64_1_6_3_1"/>
( <font color="Maroon" title="c23">o</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="c14">B</font> or <font color="Maroon" title="c23">o</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="c14">B</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_6_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:64_1_6_3_1_1"/><span class="lab"><font color="Green" title="E67">A96</font></span>: 
<font color="Maroon" title="c23">o</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="c14">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c21">y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:64_1_6_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A97</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c21">y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">o</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="E3:64_1_6_3_1_1"/>
<font color="Maroon" title="c11">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E28:64_1"><span class="lab"><font color="Green" title="E53">A40</font></span></a>, <a class="txt" href="#E30:64_1"><span class="lab"><font color="Green" title="E55">A42</font></span></a>, <a class="txt" href="#E20:64_1_6"><span class="lab"><font color="Green" title="E65">A94</font></span></a>, <a class="txt" href="#E1:64_1_6_3"><span class="lab"><font color="Green" title="E66">A95</font></span></a>, <a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</a></span>;<br/><span class="kw">hence </span><a NAME="E4:64_1_6_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c21">y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_6_3_1_1"><span class="lab"><font color="Green" title="E67">A96</font></span></a>, <a class="txt" href="#E2:64_1_6_3_1_1"><span class="lab"><font color="Green" title="E68">A97</font></span></a>, <a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</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="64_1_6_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:64_1_6_3_1_2"/><span class="lab"><font color="Green" title="E67">A98</font></span>: 
<font color="Maroon" title="c23">o</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="c14">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c21">y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:64_1_6_3_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c21">y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:64_1"><span class="lab"><font color="Green" title="E49">A8</font></span></a>, <a class="ref" href="funct_7.html#T31" title="FUNCT_7:th.31">FUNCT_7:31</a></span>;<br/><span class="kw">hence </span><a NAME="E3:64_1_6_3_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c21">y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:64_1_6"><span class="lab"><font color="Green" title="E61">A88</font></span></a>, <a class="txt" href="#E8:64_1_6"><span class="lab"><font color="Green" title="E59">A86</font></span></a>, <a class="txt" href="#E1:64_1_6_3_1_2"><span class="lab"><font color="Green" title="E67">A98</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0: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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E22:64_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c11">a1</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="c14">B</font></span>)</span>,<font color="Maroon" title="c21">y</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">L3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:64_1_6"><span class="lab"><font color="Green" title="E65">A94</font></span></a>, <a class="txt" href="#E17:64_1_6"><span class="lab"><font color="Green" title="E63">A90</font></span></a>, <a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</a></span>;<br/>
<a NAME="E23:64_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A99</font></span>: 
<font color="Maroon" title="c11">a1</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="c14">B</font></span>)</span>,<font color="Maroon" title="c21">y</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">L3</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:64_1_6"><span class="lab"><font color="Green" title="E64">A93</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c23">x1</font> = <font color="Maroon" title="c11">a1</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="c14">B</font></span>)</span>,<font color="Maroon" title="c20">x</font>);<br/>
<a NAME="E24:64_1_6"/><span class="lab"><font color="Green" title="E67">A100</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c20">x</font>)</span>)</span> <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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_6_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c24">o</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="c24">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c20">x</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:64_1_6_4"/><span class="lab"><font color="Green" title="E68">A101</font></span>: 
<font color="Maroon" title="c24">o</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c20">x</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_6_4_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="E1:64_1_6_4_1"/>
( <font color="Maroon" title="c24">o</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="c14">B</font> or <font color="Maroon" title="c24">o</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="c14">B</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_6_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:64_1_6_4_1_1"/>
<font color="Maroon" title="c24">o</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="c14">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c20">x</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:64_1_6_4_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c20">x</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">o</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/><span class="kw">hence </span><a NAME="E3:64_1_6_4_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c20">x</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E28:64_1"><span class="lab"><font color="Green" title="E53">A40</font></span></a>, <a class="txt" href="#E30:64_1"><span class="lab"><font color="Green" title="E55">A42</font></span></a>, <a class="txt" href="#E1:64_1_6_4"><span class="lab"><font color="Green" title="E68">A101</font></span></a>, <a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</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="64_1_6_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:64_1_6_4_1_2"/><span class="lab"><font color="Green" title="E69">A102</font></span>: 
<font color="Maroon" title="c24">o</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="c14">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c20">x</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:64_1_6_4_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c20">x</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:64_1"><span class="lab"><font color="Green" title="E49">A8</font></span></a>, <a class="ref" href="funct_7.html#T31" title="FUNCT_7:th.31">FUNCT_7:31</a></span>;<br/><span class="kw">hence </span><a NAME="E3:64_1_6_4_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c20">x</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:64_1"><span class="lab"><font color="Green" title="E56">A54</font></span></a>, <a class="txt" href="#E14:64_1_6"><span class="lab"><font color="Green" title="E60">A87</font></span></a>, <a class="txt" href="#E1:64_1_6_4_1_2"><span class="lab"><font color="Green" title="E69">A102</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0: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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E26:64_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A103</font></span>: 
<font color="Maroon" title="c11">a1</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="c14">B</font></span>)</span>,<font color="Maroon" title="c20">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:64_1"><span class="lab"><font color="Green" title="E53">A40</font></span></a>, <a class="txt" href="#E30:64_1"><span class="lab"><font color="Green" title="E55">A42</font></span></a>, <a class="txt" href="#E24:64_1_6"><span class="lab"><font color="Green" title="E67">A100</font></span></a>, <a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</a></span>;<br/>
<a NAME="E27:64_1_6"/><span class="lab"><font color="Green" title="E69">A104</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c20">x</font>)</span>)</span> <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="c14">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c21">y</font>)</span>)</span> <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="c14">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:64_1"><span class="lab"><font color="Green" title="E49">A8</font></span></a>, <a class="ref" href="funct_7.html#T31" title="FUNCT_7:th.31">FUNCT_7:31</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_6_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c24">o</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="c24">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">L2</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c20">x</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:64_1_6_5"/><span class="lab"><font color="Green" title="E70">A105</font></span>: 
<font color="Maroon" title="c24">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">L2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c20">x</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_6_5_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="E1:64_1_6_5_1"/>
( <font color="Maroon" title="c24">o</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="c14">B</font> or <font color="Maroon" title="c24">o</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="c14">B</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_6_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:64_1_6_5_1_1"/><span class="lab"><font color="Green" title="E71">A106</font></span>: 
<font color="Maroon" title="c24">o</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="c14">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c20">x</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:64_1_6_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A107</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c20">x</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">o</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="E3:64_1_6_5_1_1"/>
<font color="Maroon" title="c11">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E28:64_1"><span class="lab"><font color="Green" title="E53">A40</font></span></a>, <a class="txt" href="#E30:64_1"><span class="lab"><font color="Green" title="E55">A42</font></span></a>, <a class="txt" href="#E20:64_1_6"><span class="lab"><font color="Green" title="E65">A94</font></span></a>, <a class="txt" href="#E1:64_1_6_5"><span class="lab"><font color="Green" title="E70">A105</font></span></a>, <a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</a></span>;<br/><span class="kw">hence </span><a NAME="E4:64_1_6_5_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c20">x</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_6_5_1_1"><span class="lab"><font color="Green" title="E71">A106</font></span></a>, <a class="txt" href="#E2:64_1_6_5_1_1"><span class="lab"><font color="Green" title="E72">A107</font></span></a>, <a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</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="64_1_6_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:64_1_6_5_1_2"/><span class="lab"><font color="Green" title="E71">A108</font></span>: 
<font color="Maroon" title="c24">o</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="c14">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c20">x</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:64_1_6_5_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c20">x</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:64_1"><span class="lab"><font color="Green" title="E49">A8</font></span></a>, <a class="ref" href="funct_7.html#T31" title="FUNCT_7:th.31">FUNCT_7:31</a></span>;<br/><span class="kw">hence </span><a NAME="E3:64_1_6_5_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c20">x</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:64_1_6"><span class="lab"><font color="Green" title="E60">A87</font></span></a>, <a class="txt" href="#E8:64_1_6"><span class="lab"><font color="Green" title="E59">A86</font></span></a>, <a class="txt" href="#E1:64_1_6_5_1_2"><span class="lab"><font color="Green" title="E71">A108</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0: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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E29:64_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c11">a1</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="c14">B</font></span>)</span>,<font color="Maroon" title="c20">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">L3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:64_1_6"><span class="lab"><font color="Green" title="E65">A94</font></span></a>, <a class="txt" href="#E24:64_1_6"><span class="lab"><font color="Green" title="E67">A100</font></span></a>, <a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</a></span>;<br/>
<a NAME="E30:64_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c11">a1</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="c14">B</font></span>)</span>,<font color="Maroon" title="c20">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">L3</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:64_1_6"><span class="lab"><font color="Green" title="E68">A103</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E31:64_1_6"/><span class="kw">then </span>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">L3</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">S</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:64_1_6"><span class="lab"><font color="Green" title="E62">A89</font></span></a>, <a class="txt" href="#E27:64_1_6"><span class="lab"><font color="Green" title="E69">A104</font></span></a>, <a class="txt" href="#E23:64_1_6"><span class="lab"><font color="Green" title="E66">A99</font></span></a>, <a class="ref" href="pencil_1.html#T2" target="_self" title="PENCIL_1:th.2">Th2</a></span>;<br/>
<a NAME="E32:64_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A109</font></span>: 
<font color="Maroon" title="c19">L3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="pencil_1.html#D2" target="_self" title="PENCIL_1:def.2">Def2</a></span>;<br/>
<span class="kw">thus </span><a NAME="E33:64_1_6"/>
<font color="Maroon" title="c16">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">SS</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_6_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">e</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="c24">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">L</font> or <font color="Maroon" title="c24">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">SS</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c25">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E2:64_1_6_6"/><span class="lab"><font color="Green" title="E71">A110</font></span>: 
<font color="Maroon" title="c25">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">L3</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="E3:64_1_6_6"/><span class="lab"><font color="Green" title="E72">A111</font></span>: 
 ex <font color="Olive" title="b1">F</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">f</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp; (  for <font color="Olive" title="b2">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">o</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:64_1_6"><span class="lab"><font color="Green" title="E65">A94</font></span></a>, <a class="txt" href="#E2:64_1_6_6"><span class="lab"><font color="Green" title="E71">A110</font></span></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c26">f</font> = <font color="Maroon" title="c25">f</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="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/>
<span class="kw">assume </span><a NAME="E5:64_1_6_6"/><span class="lab"><font color="Green" title="E73">A112</font></span>: 
<font color="Maroon" title="c24">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">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="c24">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">SS</font></span><br/>

<div><span class="lab"><font color="Green" title="E74">A113</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_6_6_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c27">o</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="c27">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">L2</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c26">f</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c24">e</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:64_1_6_6_1"/><span class="lab"><font color="Green" title="E75">A114</font></span>: 
<font color="Maroon" title="c27">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">L2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c26">f</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c24">e</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_6_6_1_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="E1:64_1_6_6_1_1"/>
( <font color="Maroon" title="c27">o</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="c14">B</font> or <font color="Maroon" title="c27">o</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="c14">B</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_6_6_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:64_1_6_6_1_1_1"/>
<font color="Maroon" title="c27">o</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="c14">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c26">f</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c24">e</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:64_1_6_6_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c26">f</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c24">e</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">o</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/><span class="kw">hence </span><a NAME="E3:64_1_6_6_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c26">f</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c24">e</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:64_1_6"><span class="lab"><font color="Green" title="E65">A94</font></span></a>, <a class="txt" href="#E3:64_1_6_6"><span class="lab"><font color="Green" title="E72">A111</font></span></a>, <a class="txt" href="#E1:64_1_6_6_1"><span class="lab"><font color="Green" title="E75">A114</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_6_6_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:64_1_6_6_1_1_2"/><span class="lab"><font color="Green" title="E76">A115</font></span>: 
<font color="Maroon" title="c27">o</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="c14">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c26">f</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c24">e</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:64_1_6_6_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c26">f</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c24">e</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_6_6"><span class="lab"><font color="Green" title="E72">A111</font></span></a>, <a class="ref" href="funct_7.html#T31" title="FUNCT_7:th.31">FUNCT_7:31</a></span>;<br/><span class="kw">hence </span><a NAME="E3:64_1_6_6_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c26">f</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c24">e</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:64_1"><span class="lab"><font color="Green" title="E53">A40</font></span></a>, <a class="txt" href="#E5:64_1_6_6"><span class="lab"><font color="Green" title="E73">A112</font></span></a>, <a class="txt" href="#E1:64_1_6_6_1_1_2"><span class="lab"><font color="Green" title="E76">A115</font></span></a>, <a class="ref" href="funct_7.html#T31" title="FUNCT_7:th.31">FUNCT_7:31</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E7:64_1_6_6"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">f</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c24">e</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">L2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:64_1_6"><span class="lab"><font color="Green" title="E65">A94</font></span></a>, <a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E8:64_1_6_6"/><span class="kw">then </span>
<font color="Maroon" title="c26">f</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="c14">B</font></span>)</span>,<font color="Maroon" title="c24">e</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">L3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:64_1_6_6"><span class="lab"><font color="Green" title="E74">A113</font></span></a>, <a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</a></span>;<br/>
<a NAME="E9:64_1_6_6"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c26">f</font> <a href="funct_7.html#K2" title="FUNCT_7: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="c14">B</font></span>)</span>,<font color="Maroon" title="c24">e</font>)</span>)</span> <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="c14">B</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</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="c14">B</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:64_1"><span class="lab"><font color="Green" title="E53">A40</font></span></a>, <a class="txt" href="#E30:64_1"><span class="lab"><font color="Green" title="E55">A42</font></span></a>, <a class="txt" href="#E32:64_1_6"><span class="lab"><font color="Green" title="E70">A109</font></span></a>, <a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:64_1_6_6"/>
<font color="Maroon" title="c24">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">SS</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:64_1"><span class="lab"><font color="Green" title="E56">A54</font></span></a>, <a class="txt" href="#E3:64_1_6_6"><span class="lab"><font color="Green" title="E72">A111</font></span></a>, <a class="ref" href="funct_7.html#T31" title="FUNCT_7:th.31">FUNCT_7:31</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><span class="kw">end;</span></div>


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

<span class="kw">given </span><font color="Maroon" title="c4">B</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><a NAME="E3:64"/><span class="lab"><font color="Green" title="E42">A116</font></span>: 
<font color="Maroon" title="c3">S</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="c4">B</font>
 <span class="kw">and </span><br/><a NAME="E4:64"/><span class="lab"><font color="Green" title="E43">A117</font></span>: 
 for <font color="Olive" title="b1">C</font> being   <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="c4">B</font></span>)</span></span>)</span>  st <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</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="c4">B</font></span>)</span> holds <br/>( <font color="Olive" title="b1">C</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Olive" title="b1">C</font> is  <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">S</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  &amp; <font color="Maroon" title="c3">S</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Maroon" title="c3">S</font> is  <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E5:64"/>
 not <font color="Maroon" title="c3">S</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="txt" href="#E3:64"><span class="lab"><font color="Green" title="E42">A116</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"> ( <font color="Maroon" title="c3">S</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>  &amp; <font color="Maroon" title="c3">S</font> is  <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a>  )</span><br/>

<a NAME="E6:64"/><span class="lab"><font color="Green" title="E44">A118</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">B</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="E7:64"/><span class="lab"><font color="Green" title="E45">A119</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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="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">thus </span><a NAME="E8:64"/>
<font color="Maroon" title="c3">S</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</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">S</font> is  <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="pencil_1.html#D3" target="_self" title="PENCIL_1:def.3">PENCIL_1:def 3</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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font> implies <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:64_2"/><span class="lab"><font color="Green" title="E46">A120</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font>
 <span class="kw">and </span><br/><a NAME="E2:64_2"/><span class="lab"><font color="Green" title="E47">A121</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</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="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a> </span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_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="E1:64_2_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">y</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:64_2_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</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="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:64_2_1_1"/>
<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#D1" target="_self" title="PENCIL_1:def.1">Def1</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="64_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:64_2_1_2"/><span class="lab"><font color="Green" title="E48">A122</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">y</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="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a> </span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c7">y1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E3:64_2_1_2"/><span class="lab"><font color="Green" title="E49">A123</font></span>: 
<font color="Maroon" title="c7">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font>
 <span class="kw">and </span><br/><a NAME="E4:64_2_1_2"/><span class="lab"><font color="Green" title="E50">A124</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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>
 <span class="kw">and </span><br/><a NAME="E5:64_2_1_2"/><span class="lab"><font color="Green" title="E51">A125</font></span>: 
 for <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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> holds <br/><font color="Maroon" title="c7">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="Olive" title="b1">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></span>;<br/><a NAME="E6:64_2_1_2"/><span class="lab"><font color="Green" title="E52">A126</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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="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="c8">y1</font> = <font color="Maroon" title="c7">y1</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="#E4:64_2_1_2"><span class="lab"><font color="Green" title="E50">A124</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/><span class="kw">consider </span><font color="Maroon" title="c9">x1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E9:64_2_1_2"/><span class="lab"><font color="Green" title="E53">A127</font></span>: 
<font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">and </span><br/><a NAME="E10:64_2_1_2"/><span class="lab"><font color="Green" title="E54">A128</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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>
 <span class="kw">and </span><br/><a NAME="E11:64_2_1_2"/><span class="lab"><font color="Green" title="E55">A129</font></span>: 
 for <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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> holds <br/><font color="Maroon" title="c9">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="Olive" title="b1">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">x1</font> = <font color="Maroon" title="c9">x1</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="#E10:64_2_1_2"><span class="lab"><font color="Green" title="E54">A128</font></span></a>, <a class="txt" href="#E6:64_2_1_2"><span class="lab"><font color="Green" title="E52">A126</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/><div><span class="lab"><font color="Green" title="E56">A130</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:64_2_1_2_1"/><span class="lab"><font color="Green" title="E57">A131</font></span>: 
<font color="Maroon" title="c10">x1</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="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y1</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="c4">B</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2_1_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">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="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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> implies <font color="Maroon" title="c10">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:64_2_1_2_1_1"/>
<font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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 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">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_1_2_1_1_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="E1:64_2_1_2_1_1_1"/>
( <font color="Maroon" title="c11">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="c4">B</font> or <font color="Maroon" title="c11">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="c4">B</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_1_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:64_2_1_2_1_1_1_1"/>
<font color="Maroon" title="c11">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="c4">B</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="c10">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:64_2_1_2_1_1_1_1"/>
<font color="Maroon" title="c10">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64"><span class="lab"><font color="Green" title="E42">A116</font></span></a>, <a class="txt" href="#E1:64_2"><span class="lab"><font color="Green" title="E46">A120</font></span></a>, <a class="txt" href="#E2:64_2"><span class="lab"><font color="Green" title="E47">A121</font></span></a>, <a class="txt" href="#E9:64_2_1_2"><span class="lab"><font color="Green" title="E53">A127</font></span></a>, <a class="txt" href="#E3:64_2_1_2"><span class="lab"><font color="Green" title="E49">A123</font></span></a>, <a class="ref" href="pencil_1.html#T23" target="_self" title="PENCIL_1:th.23">Th23</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="64_2_1_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:64_2_1_2_1_1_1_2"/>
<font color="Maroon" title="c11">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="c4">B</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="c10">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:64_2_1_2_1_1_1_2"/>
<font color="Maroon" title="c10">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_2_1_2_1"><span class="lab"><font color="Green" title="E57">A131</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><span class="kw">hence </span><a NAME="E3:64_2_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_2_1_2"><span class="lab"><font color="Green" title="E48">A122</font></span></a>, <a class="txt" href="#E9:64_2_1_2"><span class="lab"><font color="Green" title="E53">A127</font></span></a>, <a class="txt" href="#E10:64_2_1_2"><span class="lab"><font color="Green" title="E54">A128</font></span></a>, <a class="txt" href="#E3:64_2_1_2"><span class="lab"><font color="Green" title="E49">A123</font></span></a>, <a class="txt" href="#E4:64_2_1_2"><span class="lab"><font color="Green" title="E50">A124</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</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="E14:64_2_1_2"/>
<font color="Maroon" title="c4">B</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="E15:64_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">B</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="c4">B</font></span>)</span> <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> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">B</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D2" title="PBOOLE:def.2">PBOOLE:def 2</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">C</font> = <font color="Maroon" title="c4">B</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="c4">B</font></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="c4">B</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#T2" title="YELLOW_6:th.2">YELLOW_6:2</a></span>;<br/><a NAME="E17:64_2_1_2"/><span class="lab"><font color="Green" title="E57">A132</font></span>: 
<font color="Maroon" title="c11">C</font> is  <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64"><span class="lab"><font color="Green" title="E43">A117</font></span></a></span>;<br/><a NAME="E18:64_2_1_2"/>
<font color="Maroon" title="c8">y1</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="c4">B</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">B</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:64"><span class="lab"><font color="Green" title="E45">A119</font></span></a>, <a class="txt" href="#E5:64_2_1_2"><span class="lab"><font color="Green" title="E51">A125</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">y1i</font> = <font color="Maroon" title="c8">y1</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="c4">B</font></span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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="c4">B</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#T2" title="YELLOW_6:th.2">YELLOW_6:2</a></span>;<br/><a NAME="E20:64_2_1_2"/><span class="lab"><font color="Green" title="E58">A133</font></span>: 
<font color="Maroon" title="c12">y1i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64"><span class="lab"><font color="Green" title="E42">A116</font></span></a>, <a class="txt" href="#E6:64"><span class="lab"><font color="Green" title="E44">A118</font></span></a>, <a class="txt" href="#E2:64_2"><span class="lab"><font color="Green" title="E47">A121</font></span></a>, <a class="txt" href="#E3:64_2_1_2"><span class="lab"><font color="Green" title="E49">A123</font></span></a>, <a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</a></span>;<br/><a NAME="E21:64_2_1_2"/>
<font color="Maroon" title="c10">x1</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="c4">B</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">B</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:64"><span class="lab"><font color="Green" title="E45">A119</font></span></a>, <a class="txt" href="#E11:64_2_1_2"><span class="lab"><font color="Green" title="E55">A129</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">x1i</font> = <font color="Maroon" title="c10">x1</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="c4">B</font></span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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="c4">B</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#T2" title="YELLOW_6:th.2">YELLOW_6:2</a></span>;<br/><a NAME="E23:64_2_1_2"/>
<font color="Maroon" title="c13">x1i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64"><span class="lab"><font color="Green" title="E42">A116</font></span></a>, <a class="txt" href="#E6:64"><span class="lab"><font color="Green" title="E44">A118</font></span></a>, <a class="txt" href="#E1:64_2"><span class="lab"><font color="Green" title="E46">A120</font></span></a>, <a class="txt" href="#E9:64_2_1_2"><span class="lab"><font color="Green" title="E53">A127</font></span></a>, <a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</a></span>;<br/><a NAME="E24:64_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">x1i</font>,<font color="Maroon" title="c12">y1i</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:64_2_1_2"><span class="lab"><font color="Green" title="E58">A133</font></span></a>, <a class="txt" href="#E17:64_2_1_2"><span class="lab"><font color="Green" title="E57">A132</font></span></a>, <a class="ref" href="pencil_1.html#D3" target="_self" title="PENCIL_1:def.3">Def3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">l</font> being   <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="c4">B</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E26:64_2_1_2"/><span class="lab"><font color="Green" title="E59">A134</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c13">x1i</font>,<font color="Maroon" title="c12">y1i</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:64_2_1_2"><span class="lab"><font color="Green" title="E56">A130</font></span></a>, <a class="ref" href="pencil_1.html#D1" target="_self" title="PENCIL_1:def.1">Def1</a></span>;<br/><a NAME="E27:64_2_1_2"/><span class="lab"><font color="Green" title="E60">A135</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <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="E28:64_2_1_2"/><span class="lab"><font color="Green" title="E61">A136</font></span>: 
 for <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> holds <br/><font color="Maroon" title="c8">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">a</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">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> implies <font color="Maroon" title="c8">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_2_1_2_2"/>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</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="c8">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font></span><br/>

<a NAME="E2:64_2_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A137</font></span>: 
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_1_2_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="E1:64_2_1_2_2_1"/>
( <font color="Maroon" title="c15">a</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="c4">B</font> or <font color="Maroon" title="c15">a</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="c4">B</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:64_2_1_2_2_1_1"/><span class="lab"><font color="Green" title="E63">A138</font></span>: 
<font color="Maroon" title="c15">a</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="c4">B</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="c8">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font></span><br/><div class="add"><a NAME="E2:64_2_1_2_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:64_2_1_2"><span class="lab"><font color="Green" title="E60">A135</font></span></a>, <a class="ref" href="funct_7.html#T31" title="FUNCT_7:th.31">FUNCT_7:31</a></span>;<br/><span class="kw">hence </span><a NAME="E3:64_2_1_2_2_1_1"/>
<font color="Maroon" title="c8">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:64_2_1_2"><span class="lab"><font color="Green" title="E59">A134</font></span></a>, <a class="txt" href="#E1:64_2_1_2_2_1_1"><span class="lab"><font color="Green" title="E63">A138</font></span></a>, <a class="ref" href="zfmisc_1.html#T32" title="ZFMISC_1:th.32">ZFMISC_1:32</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="64_2_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:64_2_1_2_2_1_2"/><span class="lab"><font color="Green" title="E63">A139</font></span>: 
<font color="Maroon" title="c15">a</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="c4">B</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="c8">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font></span><br/><div class="add"><a NAME="E2:64_2_1_2_2_1_2"/>
<font color="Maroon" title="c10">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</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="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:64_2_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_2_1_2_2"><span class="lab"><font color="Green" title="E62">A137</font></span></a>, <a class="ref" href="pzfmisc1.html#D1" title="PZFMISC1:def.1">PZFMISC1:def 1</a></span>;<br/><a NAME="E4:64_2_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64"><span class="lab"><font color="Green" title="E42">A116</font></span></a>, <a class="txt" href="#E1:64_2"><span class="lab"><font color="Green" title="E46">A120</font></span></a>, <a class="txt" href="#E2:64_2"><span class="lab"><font color="Green" title="E47">A121</font></span></a>, <a class="txt" href="#E9:64_2_1_2"><span class="lab"><font color="Green" title="E53">A127</font></span></a>, <a class="txt" href="#E3:64_2_1_2"><span class="lab"><font color="Green" title="E49">A123</font></span></a>, <a class="txt" href="#E1:64_2_1_2_2_1_2"><span class="lab"><font color="Green" title="E63">A139</font></span></a>, <a class="ref" href="pencil_1.html#T23" target="_self" title="PENCIL_1:th.23">Th23</a></span>;<br/><span class="kw">hence </span><a NAME="E5:64_2_1_2_2_1_2"/>
<font color="Maroon" title="c8">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_2_1_2_2_1_2"><span class="lab"><font color="Green" title="E63">A139</font></span></a>, <a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</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="E29:64_2_1_2"/>
<span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2_1_2_3"><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> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="pboole.html#D2" title="PBOOLE:def.2">PBOOLE:def 2</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="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> or <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> <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> <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="c15">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_2_1_2_3"/><span class="lab"><font color="Green" title="E62">A140</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"> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> <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> <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="c15">i</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">i1</font> = <font color="Maroon" title="c15">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> ;<br/>
<span class="kw">thus </span><a NAME="E3:64_2_1_2_3"/>
<span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> <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> <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="c15">i</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2_1_2_3_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_1_2_3_1_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="E1:64_2_1_2_3_1_1"/>
( <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="c4">B</font> or <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="c4">B</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_1_2_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:64_2_1_2_3_1_1_1"/><span class="lab"><font color="Green" title="E63">A141</font></span>: 
<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="c4">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> <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> <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="c15">i</font></span><br/><div class="add"><a NAME="E2:64_2_1_2_3_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> <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="c14">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:64_2_1_2"><span class="lab"><font color="Green" title="E60">A135</font></span></a>, <a class="ref" href="funct_7.html#T31" title="FUNCT_7:th.31">FUNCT_7:31</a></span>;<br/><a NAME="E3:64_2_1_2_3_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> <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>  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="c4">B</font></span>)</span></span>)</span>
 ;<br/><a NAME="E4:64_2_1_2_3_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> <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>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c4">B</font></span>)</span></span>)</span>
 ;<br/><span class="kw">hence </span><a NAME="E5:64_2_1_2_3_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> <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> <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="c15">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_2_1_2_3_1_1_1"><span class="lab"><font color="Green" title="E63">A141</font></span></a>, <a class="ref" href="yellow_6.html#T2" title="YELLOW_6:th.2">YELLOW_6:2</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="64_2_1_2_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:64_2_1_2_3_1_1_2"/><span class="lab"><font color="Green" title="E63">A142</font></span>: 
<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="c4">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> <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> <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="c15">i</font></span><br/><div class="add"><a NAME="E2:64_2_1_2_3_1_1_2"/>
<font color="Maroon" title="c10">x1</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> <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="c16">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:64"><span class="lab"><font color="Green" title="E45">A119</font></span></a>, <a class="txt" href="#E11:64_2_1_2"><span class="lab"><font color="Green" title="E55">A129</font></span></a></span>;<br/><a NAME="E3:64_2_1_2_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">x1</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>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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> <font color="Maroon" title="c16">i1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#T2" title="YELLOW_6:th.2">YELLOW_6:2</a></span>;<br/><a NAME="E4:64_2_1_2_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A143</font></span>: 
<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">x1</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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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> <font color="Maroon" title="c16">i1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/><span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> = 
<span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <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="#E1:64_2_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E63">A142</font></span></a>, <a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></span>
<br/>.= 
<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">x1</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="#E1:64_2_1_2_3"><span class="lab"><font color="Green" title="E62">A140</font></span></a>, <a class="ref" href="pzfmisc1.html#D1" title="PZFMISC1:def.1">PZFMISC1:def 1</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:64_2_1_2_3_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> <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> <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="c15">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_2_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E64">A143</font></span></a>, <a class="ref" href="yellow_6.html#T2" title="YELLOW_6:th.2">YELLOW_6:2</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>


</div><span class="kw">end;</span></div><a NAME="E30:64_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A144</font></span>: 
<span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>) is    <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="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></span>;<br/><a NAME="E31:64_2_1_2"/>
 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>  st <font color="Olive" title="b1">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="c4">B</font> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is 1 <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2_1_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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"> ( <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="c4">B</font> implies <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> is 1 <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_2_1_2_4"/>
<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="c4">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> is 1 <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a> </span><br/>

<span class="kw">then </span><span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> = 
<span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <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/>.= 
<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">x1</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="ref" href="pzfmisc1.html#D1" title="PZFMISC1:def.1">PZFMISC1:def 1</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:64_2_1_2_4"/>
<span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> is 1 <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a> 
 ; <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="E32:64_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A145</font></span>: 
<span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>) is  <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#D20" target="_self" title="PENCIL_1:def.20">Def20</a></span>;<br/><a NAME="E33:64_2_1_2"/><span class="lab"><font color="Green" title="E64">A146</font></span>: 
 for <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> holds <br/><font color="Maroon" title="c10">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2_1_2_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">a</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">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> implies <font color="Maroon" title="c10">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_2_1_2_5"/>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</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="c10">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font></span><br/>

<a NAME="E2:64_2_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A147</font></span>: 
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_1_2_5_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="E1:64_2_1_2_5_1"/>
( <font color="Maroon" title="c15">a</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="c4">B</font> or <font color="Maroon" title="c15">a</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="c4">B</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_1_2_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:64_2_1_2_5_1_1"/><span class="lab"><font color="Green" title="E66">A148</font></span>: 
<font color="Maroon" title="c15">a</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="c4">B</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="c10">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font></span><br/><div class="add"><a NAME="E2:64_2_1_2_5_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:64_2_1_2"><span class="lab"><font color="Green" title="E60">A135</font></span></a>, <a class="ref" href="funct_7.html#T31" title="FUNCT_7:th.31">FUNCT_7:31</a></span>;<br/><span class="kw">hence </span><a NAME="E3:64_2_1_2_5_1_1"/>
<font color="Maroon" title="c10">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:64_2_1_2"><span class="lab"><font color="Green" title="E59">A134</font></span></a>, <a class="txt" href="#E1:64_2_1_2_5_1_1"><span class="lab"><font color="Green" title="E66">A148</font></span></a>, <a class="ref" href="zfmisc_1.html#T32" title="ZFMISC_1:th.32">ZFMISC_1:32</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="64_2_1_2_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:64_2_1_2_5_1_2"/><span class="lab"><font color="Green" title="E66">A149</font></span>: 
<font color="Maroon" title="c15">a</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="c4">B</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="c10">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font></span><br/><div class="add"><a NAME="E2:64_2_1_2_5_1_2"/>
<font color="Maroon" title="c10">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</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="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:64_2_1_2_5_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_2_1_2_5"><span class="lab"><font color="Green" title="E65">A147</font></span></a>, <a class="ref" href="pzfmisc1.html#D1" title="PZFMISC1:def.1">PZFMISC1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:64_2_1_2_5_1_2"/>
<font color="Maroon" title="c10">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_2_1_2_5_1_2"><span class="lab"><font color="Green" title="E66">A149</font></span></a>, <a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</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="E34:64_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> <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="c4">B</font></span>)</span> is   <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="c4">B</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:64_2_1_2"><span class="lab"><font color="Green" title="E60">A135</font></span></a>, <a class="ref" href="funct_7.html#T31" title="FUNCT_7:th.31">FUNCT_7:31</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">L</font> =  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span> 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="#E32:64_2_1_2"><span class="lab"><font color="Green" title="E63">A145</font></span></a>, <a class="txt" href="#E30:64_2_1_2"><span class="lab"><font color="Green" title="E62">A144</font></span></a>, <a class="ref" href="pencil_1.html#D22" target="_self" title="PENCIL_1:def.22">Def22</a></span>;<br/> <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">y1</font> = 
<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 href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span>
<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="E37:64_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A150</font></span>: 
<font color="Maroon" title="c8">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:64_2_1_2"><span class="lab"><font color="Green" title="E61">A136</font></span></a>, <a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</a></span>;<br/> <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">x1</font> = 
<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 href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_7.html#K2" title="FUNCT_7: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font>)</span>)</span>
<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="E39:64_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:64_2_1_2"><span class="lab"><font color="Green" title="E64">A146</font></span></a>, <a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</a></span>;<br/><a NAME="E40:64_2_1_2"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:64_2_1_2"><span class="lab"><font color="Green" title="E53">A127</font></span></a>, <a class="txt" href="#E3:64_2_1_2"><span class="lab"><font color="Green" title="E49">A123</font></span></a>, <a class="txt" href="#E37:64_2_1_2"><span class="lab"><font color="Green" title="E65">A150</font></span></a>, <a class="ref" href="zfmisc_1.html#T32" title="ZFMISC_1:th.32">ZFMISC_1:32</a></span>;<br/><span class="kw">hence </span><a NAME="E41:64_2_1_2"/>
<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#D1" target="_self" title="PENCIL_1:def.1">Def1</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>

<span class="kw">thus </span><a NAME="E9:64"/>
<font color="Maroon" title="c3">S</font> is  <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">l</font> be   <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="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="pencil_1.html#D2" target="_self" title="PENCIL_1:def.2">PENCIL_1:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">S</font></span>)</span> implies <font color="Maroon" title="c5">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">S</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">L</font> being   <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="E2:64_3"/><span class="lab"><font color="Green" title="E46">A151</font></span>: 
<font color="Maroon" title="c5">l</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="c6">L</font>
 <span class="kw">and </span><br/><a NAME="E3:64_3"/><span class="lab"><font color="Green" title="E47">A152</font></span>: 
 ex <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> st <font color="Maroon" title="c6">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is   <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> <font color="Olive" title="b1">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#D22" target="_self" title="PENCIL_1:def.22">Def22</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:64_3"/><span class="lab"><font color="Green" title="E48">A153</font></span>: 
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">S</font></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="c5">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">S</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:64_3"/><span class="lab"><font color="Green" title="E49">A154</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">l</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">S</font>
 <span class="kw">and </span><br/><a NAME="E7:64_3"/><span class="lab"><font color="Green" title="E50">A155</font></span>: 
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">l</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">S</font>
 <span class="kw">and </span><br/><a NAME="E8:64_3"/><span class="lab"><font color="Green" title="E51">A156</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#T2" target="_self" title="PENCIL_1:th.2">Th2</a></span>;<br/>
<a NAME="E9:64_3"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">S</font></span>)</span> <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="c5">l</font>
 
<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>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E10:64_3"/><span class="kw">then </span>
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="c5">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_3"><span class="lab"><font color="Green" title="E48">A153</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">L</font> = <font color="Maroon" title="c6">L</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="txt" href="#E2:64_3"><span class="lab"><font color="Green" title="E46">A151</font></span></a>, <a class="ref" href="pencil_1.html#T13" target="_self" title="PENCIL_1:th.13">Th13</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">a1</font> = <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c11">b1</font> = <font color="Maroon" title="c8">b</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="#E6:64_3"><span class="lab"><font color="Green" title="E49">A154</font></span></a>, <a class="txt" href="#E7:64_3"><span class="lab"><font color="Green" title="E50">A155</font></span></a>, <a class="ref" href="pencil_1.html#T14" target="_self" title="PENCIL_1:th.14">Th14</a></span>;<br/>
<span class="lab"><font color="Green" title="E52">A157</font></span>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">L</font> = 
<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 href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">B</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="E14:64_3"/><span class="lab"><font color="Green" title="E53">A158</font></span>: 
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">B</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="c9">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64"><span class="lab"><font color="Green" title="E42">A116</font></span></a>, <a class="txt" href="#E4:64_3"><span class="lab"><font color="Green" title="E48">A153</font></span></a>, <a class="txt" href="#E2:64_3"><span class="lab"><font color="Green" title="E46">A151</font></span></a>, <a class="ref" href="pencil_1.html#T26" target="_self" title="PENCIL_1:th.26">Th26</a></span>;<br/>
<a NAME="E15:64_3"/>
 for <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">L</font> holds <br/><font color="Maroon" title="c9">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">a</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="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">L</font> implies <font color="Maroon" title="c9">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_3_2"/>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">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="c9">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">a</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_3_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="E1:64_3_2_1"/>
( <font color="Maroon" title="c12">a</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="c4">B</font> or <font color="Maroon" title="c12">a</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="c4">B</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_3_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:64_3_2_1_1"/>
<font color="Maroon" title="c12">a</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="c4">B</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">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">a</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:64_3_2_1_1"/>
<font color="Maroon" title="c9">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64"><span class="lab"><font color="Green" title="E42">A116</font></span></a>, <a class="txt" href="#E4:64_3"><span class="lab"><font color="Green" title="E48">A153</font></span></a>, <a class="txt" href="#E2:64_3"><span class="lab"><font color="Green" title="E46">A151</font></span></a>, <a class="ref" href="pencil_1.html#T26" target="_self" title="PENCIL_1:th.26">Th26</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="64_3_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:64_3_2_1_2"/><span class="lab"><font color="Green" title="E54">A159</font></span>: 
<font color="Maroon" title="c12">a</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="c4">B</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">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">a</font></span><br/><div class="add"><a NAME="E2:64_3_2_1_2"/>
<font color="Maroon" title="c4">B</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="E3:64_3_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">B</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="c4">B</font></span>)</span> <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> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">B</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D2" title="PBOOLE:def.2">PBOOLE:def 2</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">C</font> = <font color="Maroon" title="c4">B</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="c4">B</font></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="c4">B</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#T2" title="YELLOW_6:th.2">YELLOW_6:2</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c14">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font><span class="kw"> such that </span><br/><a NAME="E6:64_3_2_1_2"/><span class="lab"><font color="Green" title="E55">A160</font></span>: 
<font color="Maroon" title="c9">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> is   <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> <font color="Maroon" title="c14">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_3"><span class="lab"><font color="Green" title="E47">A152</font></span></a></span>;<br/><a NAME="E7:64_3_2_1_2"/>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:64_3_2_1_2"><span class="lab"><font color="Green" title="E55">A160</font></span></a>, <a class="ref" href="pencil_1.html#D6" target="_self" title="PENCIL_1:def.6">Def6</a></span>;<br/><a NAME="E8:64_3_2_1_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c9">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</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" target="_self" title="PENCIL_1:th.4">Th4</a></span>;<br/><a NAME="E9:64_3_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">j</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="c9">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#D21" target="_self" title="PENCIL_1:def.21">Def21</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">Li</font> = <font color="Maroon" title="c9">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="c4">B</font></span>)</span> as   <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="c4">B</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:64_3"><span class="lab"><font color="Green" title="E53">A158</font></span></a>, <a class="txt" href="#E6:64_3_2_1_2"><span class="lab"><font color="Green" title="E55">A160</font></span></a></span>;<br/><a NAME="E11:64_3_2_1_2"/><span class="lab"><font color="Green" title="E56">A161</font></span>: 
<font color="Maroon" title="c13">C</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="txt" href="#E4:64"><span class="lab"><font color="Green" title="E43">A117</font></span></a></span>;<br/><a NAME="E12:64_3_2_1_2"/>
<font color="Maroon" title="c10">a1</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="c9">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:64_3"><span class="lab"><font color="Green" title="E49">A154</font></span></a>, <a class="txt" href="#E2:64_3"><span class="lab"><font color="Green" title="E46">A151</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E13:64_3_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A162</font></span>: 
<font color="Maroon" title="c10">a1</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="c4">B</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">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="c4">B</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:64"><span class="lab"><font color="Green" title="E44">A118</font></span></a>, <a class="txt" href="#E13:64_3"><span class="lab"><font color="Green" title="E52">A157</font></span></a>, <a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</a></span>;<br/><a NAME="E14:64_3_2_1_2"/>
<font color="Maroon" title="c11">b1</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="c9">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:64_3"><span class="lab"><font color="Green" title="E50">A155</font></span></a>, <a class="txt" href="#E2:64_3"><span class="lab"><font color="Green" title="E46">A151</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E15:64_3_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A163</font></span>: 
<font color="Maroon" title="c11">b1</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="c4">B</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">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="c4">B</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:64"><span class="lab"><font color="Green" title="E44">A118</font></span></a>, <a class="txt" href="#E13:64_3"><span class="lab"><font color="Green" title="E52">A157</font></span></a>, <a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</a></span>;<br/><a NAME="E16:64_3_2_1_2"/><span class="lab"><font color="Green" title="E59">A164</font></span>: 
<font color="Maroon" title="c11">b1</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="c4">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64"><span class="lab"><font color="Green" title="E42">A116</font></span></a>, <a class="txt" href="#E7:64_3"><span class="lab"><font color="Green" title="E50">A155</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E17:64_3_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">b1</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="c4">B</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</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="c4">B</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:64"><span class="lab"><font color="Green" title="E44">A118</font></span></a>, <a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</a></span>;<br/><a NAME="E18:64_3_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A165</font></span>: 
<font color="Maroon" title="c11">b1</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="c4">B</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Li</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:64_3_2_1_2"><span class="lab"><font color="Green" title="E58">A163</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E19:64_3_2_1_2"/><span class="lab"><font color="Green" title="E61">A166</font></span>: 
<font color="Maroon" title="c10">a1</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="c4">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64"><span class="lab"><font color="Green" title="E42">A116</font></span></a>, <a class="txt" href="#E6:64_3"><span class="lab"><font color="Green" title="E49">A154</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><div><span class="lab"><font color="Green" title="E62">A167</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_3_2_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:64_3_2_1_2_1"/><span class="lab"><font color="Green" title="E63">A168</font></span>: 
<font color="Maroon" title="c10">a1</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="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b1</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="c4">B</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:64_3_2_1_2_1"/><span class="lab"><font color="Green" title="E64">A169</font></span>: 
 for <font color="Olive" title="b1">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">a1</font> holds <br/><font color="Maroon" title="c10">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">o</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_3_2_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">o</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="c16">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">a1</font> implies <font color="Maroon" title="c10">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">o</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_3_2_1_2_1_1"/>
<font color="Maroon" title="c16">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">a1</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="c10">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">o</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_3_2_1_2_1_1_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="E1:64_3_2_1_2_1_1_1"/>
( <font color="Maroon" title="c16">o</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="c4">B</font> or <font color="Maroon" title="c16">o</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="c4">B</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_3_2_1_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:64_3_2_1_2_1_1_1_1"/>
<font color="Maroon" title="c16">o</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="c4">B</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="c10">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">o</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:64_3_2_1_2_1_1_1_1"/>
<font color="Maroon" title="c10">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:64_3_2_1_2"><span class="lab"><font color="Green" title="E61">A166</font></span></a>, <a class="txt" href="#E16:64_3_2_1_2"><span class="lab"><font color="Green" title="E59">A164</font></span></a>, <a class="ref" href="pencil_1.html#T23" target="_self" title="PENCIL_1:th.23">Th23</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="64_3_2_1_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:64_3_2_1_2_1_1_1_2"/>
<font color="Maroon" title="c16">o</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="c4">B</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="c10">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">o</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:64_3_2_1_2_1_1_1_2"/>
<font color="Maroon" title="c10">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_3_2_1_2_1"><span class="lab"><font color="Green" title="E63">A168</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 href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">a1</font> = 
<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 href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">b1</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">hence </span><a NAME="E4:64_3_2_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:64_3"><span class="lab"><font color="Green" title="E51">A156</font></span></a>, <a class="txt" href="#E2:64_3_2_1_2_1"><span class="lab"><font color="Green" title="E64">A169</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</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="E21:64_3_2_1_2"/>
<font color="Maroon" title="c10">a1</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="c4">B</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</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="c4">B</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:64"><span class="lab"><font color="Green" title="E44">A118</font></span></a>, <a class="txt" href="#E19:64_3_2_1_2"><span class="lab"><font color="Green" title="E61">A166</font></span></a>, <a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</a></span>;<br/><a NAME="E22:64_3_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">a1</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="c4">B</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Li</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:64_3_2_1_2"><span class="lab"><font color="Green" title="E57">A162</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E23:64_3_2_1_2"/><span class="kw">then </span>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Li</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">C</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:64_3_2_1_2"><span class="lab"><font color="Green" title="E62">A167</font></span></a>, <a class="txt" href="#E18:64_3_2_1_2"><span class="lab"><font color="Green" title="E60">A165</font></span></a>, <a class="ref" href="pencil_1.html#T2" target="_self" title="PENCIL_1:th.2">Th2</a></span>;<br/><span class="kw">hence </span><a NAME="E24:64_3_2_1_2"/>
<font color="Maroon" title="c9">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_3_2_1_2"><span class="lab"><font color="Green" title="E54">A159</font></span></a>, <a class="txt" href="#E11:64_3_2_1_2"><span class="lab"><font color="Green" title="E56">A161</font></span></a>, <a class="ref" href="pencil_1.html#D2" target="_self" title="PENCIL_1:def.2">Def2</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>
<span class="kw">hence </span><a NAME="E16:64_3"/>
<font color="Maroon" title="c5">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64"><span class="lab"><font color="Green" title="E42">A116</font></span></a>, <a class="txt" href="#E2:64_3"><span class="lab"><font color="Green" title="E46">A151</font></span></a>, <a class="txt" href="#E13:64_3"><span class="lab"><font color="Green" title="E52">A157</font></span></a>, <a class="ref" href="card_3.html#T27" title="CARD_3:th.27">CARD_3:27</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>
