<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">I</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being <a href="pencil_1.html#V15" title="PENCIL_1:attr.15">PLS-yielding</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><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#K6" title="PENCIL_1:func.6">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#M4" title="PBOOLE:mode.4">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/><b>let </b><font color="Maroon" title="c2">A</font> be  <a href="pencil_1.html#V15" title="PENCIL_1:attr.15">PLS-yielding</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">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#K6" title="PENCIL_1:func.6">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#M4" title="PBOOLE:mode.4">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/><b>let </b><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#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( 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#M4" title="PBOOLE:mode.4">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="E1:66"/><i><font color="Green" title="E44">A1</font></i>: 
 <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>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>thus </b><a NAME="E2:66"/>
( 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#M4" title="PBOOLE:mode.4">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:()"><i><font color="Red">::  thesis: </font></i></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#M4" title="PBOOLE:mode.4">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="66_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:66_1"/><i><font color="Green" title="E45">A2</font></i>: 
( 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:()"><i><font color="Red">::  thesis: </font></i></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#M4" title="PBOOLE:mode.4">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="E2:66_1"/><b>then </b>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c3">S</font>
 
<b>by </b><i><a class="ref" href="pencil_1.html#T4" target="_self" title="PENCIL_1:th.4">Th4</a></i>;<br/>
<b>then consider </b><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> <b> such that </b><br/><a NAME="E4:66_1"/><i><font color="Green" title="E46">A3</font></i>: 
( <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> &amp; <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> &amp; <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> )
 <b>by </b><i><a class="ref" href="pencil_1.html#T2" target="_self" title="PENCIL_1:th.2">Th2</a></i>;<br/>
<b>reconsider </b><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#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> <b>by </b><i><a class="txt" href="#E4:66_1"><i><font color="Green" title="E46">A3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">a1</font> = <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c9">b1</font> = <font color="Maroon" title="c7">b</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="ref" href="pencil_1.html#T14" target="_self" title="PENCIL_1:th.14">Th14</a></i>;<br/>
<a NAME="E7:66_1"/><i><font color="Green" title="E47">A4</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E8:66_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> 
 
<b>by </b><i><a class="txt" href="#E1:66_1"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="#E4:66_1"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="pencil_1.html#D3" target="_self" title="PENCIL_1:def.3">Def3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">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#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span><b> such that </b><br/><a NAME="E10:66_1"/><i><font color="Green" title="E48">A5</font></i>: 
<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="c10">C</font>
 <b>by </b><i><a class="txt" href="#E4:66_1"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="pencil_1.html#D1" target="_self" title="PENCIL_1:def.1">Def1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">CC</font> being  <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>  <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E12:66_1"/><i><font color="Green" title="E49">A6</font></i>: 
( <font color="Maroon" title="c10">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="c11">CC</font> &amp;  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="c11">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> )
 <b>by </b><i><a class="ref" href="pencil_1.html#D22" target="_self" title="PENCIL_1:def.22">Def22</a></i>;<br/>
<a NAME="E13:66_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="c11">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="c11">CC</font> )
 
<b>by </b><i><a class="txt" href="#E10:66_1"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="#E12:66_1"><i><font color="Green" title="E49">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#T38" title="ZFMISC_1:th.38">ZFMISC_1:38</a></i>;<br/>
<a NAME="E14:66_1"/><b>then </b>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c11">CC</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:66_1"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="pencil_1.html#T2" target="_self" title="PENCIL_1:th.2">Th2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">CC</font> = <font color="Maroon" title="c11">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#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="ref" href="pencil_1.html#T13" target="_self" title="PENCIL_1:th.13">Th13</a></i>;<br/>
<a NAME="E16:66_1"/><i><font color="Green" title="E50">A7</font></i>: 
( <font color="Maroon" title="c8">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="c12">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="c12">CC</font></span>)</span> &amp; <font color="Maroon" title="c9">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="c12">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="c12">CC</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E4:66_1"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="card_3.html#D6" title="CARD_3:def.6">CARD_3:def 6</a></i>;<br/>
<div><i><font color="Green" title="E51">A8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_1_1"/><i><font color="Green" title="E52">A9</font></i>: 
<font color="Maroon" title="c8">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="c12">CC</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">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="c12">CC</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:66_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="c8">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="c9">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="66_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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="c8">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="c9">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> )</span><br/>

<b>assume </b><a NAME="E1:66_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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">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="c9">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="66_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_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="c12">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="c12">CC</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:66_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="c12">CC</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">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="c9">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"><b>hence </b><a NAME="E2:66_1_1_1_1_1"/>
<font color="Maroon" title="c8">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="c9">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font>
 <b>by </b><i><a class="txt" href="#E1:66_1_1"><i><font color="Green" title="E52">A9</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:66_1_1_1_1_2"/><i><font color="Green" title="E53">A10</font></i>: 
<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="c12">CC</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">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="c9">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="E2:66_1_1_1_1_2"/>
( <font color="Maroon" title="c8">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="c12">CC</font> &amp; <font color="Maroon" title="c9">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="c12">CC</font> )
 <b>by </b><i><a class="txt" href="#E10:66_1"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="#E12:66_1"><i><font color="Green" title="E49">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#T38" title="ZFMISC_1:th.38">ZFMISC_1:38</a></i>;<br/><b>hence </b><a NAME="E3:66_1_1_1_1_2"/>
<font color="Maroon" title="c8">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="c9">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font>
 <b>by </b><i><a class="txt" href="#E1:66_1_1_1_1_2"><i><font color="Green" title="E53">A10</font></i></a>, <a class="ref" href="pencil_1.html#T23" target="_self" title="PENCIL_1:th.23">Th23</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E3:66_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E4:66_1"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="#E7:66_1"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E18:66_1"/><b>then </b>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><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="c12">CC</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E16:66_1"><i><font color="Green" title="E50">A7</font></i></a>, <a class="ref" href="pencil_1.html#T2" target="_self" title="PENCIL_1:th.2">Th2</a></i>;<br/>
<b>then reconsider </b><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="c12">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>  <b>by </b><i><a class="ref" href="pencil_1.html#T4" target="_self" title="PENCIL_1:th.4">Th4</a></i>;<br/>
<a NAME="E20:66_1"/>
<font color="Maroon" title="c12">CC</font> <a href="polynom1.html#K1" title="POLYNOM1: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="c12">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="66_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="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="c12">CC</font> <a href="polynom1.html#K1" title="POLYNOM1: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="c12">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/>

<b>assume </b><a NAME="E1:66_1_2"/><i><font color="Green" title="E52">A11</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c12">CC</font> <a href="polynom1.html#K1" title="POLYNOM1: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="c12">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="66_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_2_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="c12">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="c12">CC</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_2_1_1"><b>suppose </b></a><a NAME="E1:66_1_2_1_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="c12">CC</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c12">CC</font> <a href="polynom1.html#K1" title="POLYNOM1: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="c12">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="E2:66_1_2_1_1"/><b>then </b><i><font color="Green" title="E53">A12</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">CC</font> <a href="polynom1.html#K1" title="POLYNOM1: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="c12">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="c12">CC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><a NAME="E3:66_1_2_1_1"/>
<font color="Maroon" title="c12">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>
 <b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/><b>hence </b><a NAME="E4:66_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">CC</font> <a href="polynom1.html#K1" title="POLYNOM1: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="c12">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>
 <b>by </b><i><a class="txt" href="#E1:66_1_2"><i><font color="Green" title="E52">A11</font></i></a>, <a class="txt" href="#E2:66_1_2_1_1"><i><font color="Green" title="E53">A12</font></i></a>, <a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_2_1_2"><b>suppose </b></a><a NAME="E1:66_1_2_1_2"/><i><font color="Green" title="E53">A13</font></i>: 
<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="c12">CC</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c12">CC</font> <a href="polynom1.html#K1" title="POLYNOM1: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="c12">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="E2:66_1_2_1_2"/><i><font color="Green" title="E54">A14</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">CC</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:66_1_2_1_2"/>
<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="66_1_2_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="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/>

<b>assume </b><a NAME="E1:66_1_2_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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>then consider </b><font color="Maroon" title="c16">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:66_1_2_1_2_1"/><i><font color="Green" title="E55">A15</font></i>: 
( <font color="Maroon" title="c16">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="Maroon" title="c16">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> )
 <b>by </b><i><a class="txt" href="#E1:66_1_2_1_2"><i><font color="Green" title="E53">A13</font></i></a>, <a class="ref" href="card_3.html#D6" title="CARD_3:def.6">CARD_3:def 6</a></i>;<br/>
<a NAME="E4:66_1_2_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>
 
<b>by </b><i><a class="txt" href="#E1:66_1_2"><i><font color="Green" title="E52">A11</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:66_1_2_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>
 <b>by </b><i><a class="txt" href="#E3:66_1_2_1_2_1"><i><font color="Green" title="E55">A15</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:66_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">CC</font> <a href="polynom1.html#K1" title="POLYNOM1: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="c12">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>
 <b>by </b><i><a class="txt" href="#E1:66_1_2_1_2"><i><font color="Green" title="E53">A13</font></i></a>, <a class="txt" href="#E2:66_1_2_1_2"><i><font color="Green" title="E54">A14</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c14">B</font> = <font color="Maroon" title="c12">CC</font> <a href="polynom1.html#K1" title="POLYNOM1: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="c12">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#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="ref" href="pencil_1.html#T27" target="_self" title="PENCIL_1:th.27">Th27</a>, <a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E22:66_1"/><i><font color="Green" title="E52">A16</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">CC</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E23:66_1"/><b>then </b><i><font color="Green" title="E53">A17</font></i>: 
<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="c12">CC</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font>
 
<b>by </b><i><a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/>
<a NAME="E24:66_1"/><i><font color="Green" title="E54">A18</font></i>: 
 <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>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c14">B</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="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="E25:66_1"/><i><font color="Green" title="E55">A19</font></i>: 
for <font color="Olive" title="b1">f</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.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="c12">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="c12">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="66_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">f</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">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="c12">CC</font> holds <br/><font color="Maroon" title="c15">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="c12">CC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> )</span><br/>

<b>assume </b><a NAME="E1:66_1_3"/><i><font color="Green" title="E56">A20</font></i>: 
<font color="Maroon" title="c15">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>  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="c12">CC</font> holds <br/><font color="Maroon" title="c15">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="c12">CC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c16">f1</font> = <font color="Maroon" title="c15">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#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> ;<br/>
<b>let </b><font color="Maroon" title="c17">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c17">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="c12">CC</font> implies <font color="Maroon" title="c15">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="c12">CC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font> )</span><br/>

<b>assume </b><a NAME="E3:66_1_3"/><i><font color="Green" title="E57">A21</font></i>: 
<font color="Maroon" title="c17">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="c12">CC</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c15">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="c12">CC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_1_3_1"/><i><font color="Green" title="E58">A22</font></i>: 
not <font color="Maroon" title="c15">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="c12">CC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:66_1_3_1"/>
<font color="Maroon" title="c8">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="c12">CC</font>
 <b>by </b><i><a class="txt" href="#E10:66_1"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="#E12:66_1"><i><font color="Green" title="E49">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#T38" title="ZFMISC_1:th.38">ZFMISC_1:38</a></i>;<br/><a NAME="E3:66_1_3_1"/><b>then </b><i><font color="Green" title="E59">A23</font></i>: 
<font color="Maroon" title="c8">a1</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="c12">CC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font>
 <b>by </b><i><a class="txt" href="#E22:66_1"><i><font color="Green" title="E52">A16</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><a NAME="E4:66_1_3_1"/>
<font color="Maroon" title="c9">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="c12">CC</font>
 <b>by </b><i><a class="txt" href="#E10:66_1"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="#E12:66_1"><i><font color="Green" title="E49">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#T38" title="ZFMISC_1:th.38">ZFMISC_1:38</a></i>;<br/><a NAME="E5:66_1_3_1"/><b>then </b><i><font color="Green" title="E60">A24</font></i>: 
<font color="Maroon" title="c9">b1</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="c12">CC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font>
 <b>by </b><i><a class="txt" href="#E22:66_1"><i><font color="Green" title="E52">A16</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><a NAME="E6:66_1_3_1"/>
<font color="Maroon" title="c16">f1</font>,<font color="Maroon" title="c6">a</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a> 
 <b>by </b><i><a class="txt" href="#E1:66_1"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="#E4:66_1"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="#E1:66_1_3"><i><font color="Green" title="E56">A20</font></i></a>, <a class="ref" href="pencil_1.html#D3" target="_self" title="PENCIL_1:def.3">Def3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c18">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#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span><b> such that </b><br/><a NAME="E8:66_1_3_1"/><i><font color="Green" title="E61">A25</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c16">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="c18">la</font>
 <b>by </b><i><a class="txt" href="#E1:66_1_3_1"><i><font color="Green" title="E58">A22</font></i></a>, <a class="txt" href="#E3:66_1_3_1"><i><font color="Green" title="E59">A23</font></i></a>, <a class="ref" href="pencil_1.html#D1" target="_self" title="PENCIL_1:def.1">Def1</a></i>;<br/><b>consider </b><font color="Maroon" title="c19">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#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E10:66_1_3_1"/><i><font color="Green" title="E62">A26</font></i>: 
( <font color="Maroon" title="c18">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="c19">La</font> &amp; <font color="Maroon" title="c19">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="c19">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="c19">La</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_1.html#T24" target="_self" title="PENCIL_1:th.24">Th24</a></i>;<br/><a NAME="E11:66_1_3_1"/><i><font color="Green" title="E63">A27</font></i>: 
( <font color="Maroon" title="c15">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="c19">La</font> &amp; <font color="Maroon" title="c8">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="c19">La</font> )
 <b>by </b><i><a class="txt" href="#E8:66_1_3_1"><i><font color="Green" title="E61">A25</font></i></a>, <a class="txt" href="#E10:66_1_3_1"><i><font color="Green" title="E62">A26</font></i></a>, <a class="ref" href="zfmisc_1.html#T38" title="ZFMISC_1:th.38">ZFMISC_1:38</a></i>;<br/><a NAME="E12:66_1_3_1"/><b>then </b>
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c19">La</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i</font>
 <b>by </b><i><a class="txt" href="#E1:66_1_3_1"><i><font color="Green" title="E58">A22</font></i></a>, <a class="txt" href="#E3:66_1_3_1"><i><font color="Green" title="E59">A23</font></i></a>, <a class="ref" href="pencil_1.html#T23" target="_self" title="PENCIL_1:th.23">Th23</a></i>;<br/><a NAME="E13:66_1_3_1"/><b>then </b>
( not <font color="Maroon" title="c19">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="c12">CC</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c19">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="c12">CC</font></span>)</span> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> )
 <b>by </b><i><a class="txt" href="#E3:66_1_3"><i><font color="Green" title="E57">A21</font></i></a>, <a class="ref" href="pencil_1.html#T12" target="_self" title="PENCIL_1:th.12">Th12</a></i>;<br/><b>then consider </b><font color="Maroon" title="c20">ca</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E15:66_1_3_1"/><i><font color="Green" title="E64">A28</font></i>: 
<font color="Maroon" title="c19">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="c12">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="c20">ca</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="realset1.html#D4" title="REALSET1:def.4">REALSET1:def 4</a></i>;<br/><a NAME="E16:66_1_3_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c19">La</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E17:66_1_3_1"/><b>then </b><i><font color="Green" title="E65">A29</font></i>: 
( <font color="Maroon" title="c8">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="c12">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="c20">ca</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">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="c12">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="c20">ca</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="#E11:66_1_3_1"><i><font color="Green" title="E63">A27</font></i></a>, <a class="txt" href="#E15:66_1_3_1"><i><font color="Green" title="E64">A28</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><a NAME="E18:66_1_3_1"/>
<font color="Maroon" title="c16">f1</font>,<font color="Maroon" title="c7">b</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a> 
 <b>by </b><i><a class="txt" href="#E1:66_1"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="#E4:66_1"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="#E1:66_1_3"><i><font color="Green" title="E56">A20</font></i></a>, <a class="ref" href="pencil_1.html#D3" target="_self" title="PENCIL_1:def.3">Def3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c21">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#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span><b> such that </b><br/><a NAME="E20:66_1_3_1"/><i><font color="Green" title="E66">A30</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c16">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="c21">lb</font>
 <b>by </b><i><a class="txt" href="#E1:66_1_3_1"><i><font color="Green" title="E58">A22</font></i></a>, <a class="txt" href="#E5:66_1_3_1"><i><font color="Green" title="E60">A24</font></i></a>, <a class="ref" href="pencil_1.html#D1" target="_self" title="PENCIL_1:def.1">Def1</a></i>;<br/><b>consider </b><font color="Maroon" title="c22">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#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E22:66_1_3_1"/><i><font color="Green" title="E67">A31</font></i>: 
( <font color="Maroon" title="c21">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="c22">Lb</font> &amp; <font color="Maroon" title="c22">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="c22">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="c22">Lb</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_1.html#T24" target="_self" title="PENCIL_1:th.24">Th24</a></i>;<br/><a NAME="E23:66_1_3_1"/><i><font color="Green" title="E68">A32</font></i>: 
( <font color="Maroon" title="c15">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="c22">Lb</font> &amp; <font color="Maroon" title="c9">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="c22">Lb</font> )
 <b>by </b><i><a class="txt" href="#E20:66_1_3_1"><i><font color="Green" title="E66">A30</font></i></a>, <a class="txt" href="#E22:66_1_3_1"><i><font color="Green" title="E67">A31</font></i></a>, <a class="ref" href="zfmisc_1.html#T38" title="ZFMISC_1:th.38">ZFMISC_1:38</a></i>;<br/><a NAME="E24:66_1_3_1"/><b>then </b>
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c22">Lb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i</font>
 <b>by </b><i><a class="txt" href="#E1:66_1_3_1"><i><font color="Green" title="E58">A22</font></i></a>, <a class="txt" href="#E5:66_1_3_1"><i><font color="Green" title="E60">A24</font></i></a>, <a class="ref" href="pencil_1.html#T23" target="_self" title="PENCIL_1:th.23">Th23</a></i>;<br/><a NAME="E25:66_1_3_1"/><b>then </b>
( not <font color="Maroon" title="c22">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="c12">CC</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c22">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="c12">CC</font></span>)</span> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> )
 <b>by </b><i><a class="txt" href="#E3:66_1_3"><i><font color="Green" title="E57">A21</font></i></a>, <a class="ref" href="pencil_1.html#T12" target="_self" title="PENCIL_1:th.12">Th12</a></i>;<br/><b>then consider </b><font color="Maroon" title="c23">cb</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E27:66_1_3_1"/><i><font color="Green" title="E69">A33</font></i>: 
<font color="Maroon" title="c22">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="c12">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="c23">cb</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="realset1.html#D4" title="REALSET1:def.4">REALSET1:def 4</a></i>;<br/><a NAME="E28:66_1_3_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c22">Lb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E29:66_1_3_1"/><b>then </b><i><font color="Green" title="E70">A34</font></i>: 
( <font color="Maroon" title="c9">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="c12">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="c23">cb</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">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="c12">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="c23">cb</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="#E23:66_1_3_1"><i><font color="Green" title="E68">A32</font></i></a>, <a class="txt" href="#E27:66_1_3_1"><i><font color="Green" title="E69">A33</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><a NAME="E30:66_1_3_1"/><b>then </b>
<font color="Maroon" title="c9">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="c12">CC</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">cb</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E31:66_1_3_1"/><b>then </b>
<font color="Maroon" title="c20">ca</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">cb</font>
 <b>by </b><i><a class="txt" href="#E17:66_1"><i><font color="Green" title="E51">A8</font></i></a>, <a class="txt" href="#E17:66_1_3_1"><i><font color="Green" title="E65">A29</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E32:66_1_3_1"/><b>then </b>
<font color="Maroon" title="c15">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="c12">CC</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">cb</font>
 <b>by </b><i><a class="txt" href="#E17:66_1_3_1"><i><font color="Green" title="E65">A29</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E33:66_1_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E29:66_1_3_1"><i><font color="Green" title="E70">A34</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:66_1_3"/>
<font color="Maroon" title="c15">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="c12">CC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E26:66_1"/><i><font color="Green" title="E56">A35</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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="66_1_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:66_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>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="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="66_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="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/>

<b>assume </b><a NAME="E1:66_1_4_1"/><i><font color="Green" title="E57">A36</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>then reconsider </b><font color="Maroon" title="c16">f</font> = <font color="Maroon" title="c15">e</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="ref" href="pencil_1.html#T14" target="_self" title="PENCIL_1:th.14">Th14</a></i>;<br/>
<a NAME="E3:66_1_4_1"/><i><font color="Green" title="E58">A37</font></i>: 
 <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>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_4_1_1"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>assume </b><a NAME="E1:66_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:()"><i><font color="Red">::  thesis: </font></i></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/><b>then reconsider </b><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="66_1_4_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_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="c12">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="c12">CC</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_4_1_1_1_1"><b>suppose </b></a><a NAME="E1:66_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="c12">CC</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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"><b>hence </b><a NAME="E2:66_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>
 <b>by </b><i><a class="txt" href="#E23:66_1"><i><font color="Green" title="E53">A17</font></i></a>, <a class="txt" href="#E1:66_1_4_1"><i><font color="Green" title="E57">A36</font></i></a>, <a class="ref" href="card_3.html#D6" title="CARD_3:def.6">CARD_3:def 6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_4_1_1_1_2"><b>suppose </b></a><a NAME="E1:66_1_4_1_1_1_2"/><i><font color="Green" title="E59">A38</font></i>: 
<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="c12">CC</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="E2:66_1_4_1_1_1_2"/><b>then </b>
<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="c12">CC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font>
 <b>by </b><i><a class="txt" href="#E25:66_1"><i><font color="Green" title="E55">A19</font></i></a>, <a class="txt" href="#E1:66_1_4_1"><i><font color="Green" title="E57">A36</font></i></a></i>;<br/><b>hence </b><a NAME="E3:66_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>
 <b>by </b><i><a class="txt" href="#E1:66_1_4_1_1_1_2"><i><font color="Green" title="E59">A38</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:66_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>
 <b>by </b><i><a class="txt" href="#E24:66_1"><i><font color="Green" title="E54">A18</font></i></a>, <a class="txt" href="#E3:66_1_4_1"><i><font color="Green" title="E58">A37</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>let </b><font color="Maroon" title="c15">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="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/>

<b>assume </b><a NAME="E2:66_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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>then consider </b><font color="Maroon" title="c16">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:66_1_4"/><i><font color="Green" title="E57">A39</font></i>: 
( <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> &amp; ( 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> ) )
 <b>by </b><i><a class="txt" href="#E24:66_1"><i><font color="Green" title="E54">A18</font></i></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>
<a NAME="E5:66_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="c12">CC</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">p</font>
 
<b>by </b><i><a class="txt" href="#E23:66_1"><i><font color="Green" title="E53">A17</font></i></a>, <a class="txt" href="#E4:66_1_4"><i><font color="Green" title="E57">A39</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E7:66_1_4"/><i><font color="Green" title="E58">A40</font></i>: 
( <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> &amp; <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="c12">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="c12">CC</font></span>)</span> )
 <b>by </b><i><a class="ref" href="card_3.html#D6" title="CARD_3:def.6">CARD_3:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">g</font> = <font color="Maroon" title="c17">g</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="txt" href="#E7:66_1_4"><i><font color="Green" title="E58">A40</font></i></a>, <a class="ref" href="pencil_1.html#T14" target="_self" title="PENCIL_1:th.14">Th14</a></i>;<br/>
<a NAME="E9:66_1_4"/><i><font color="Green" title="E59">A41</font></i>: 
 <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>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_4_2"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>assume </b><a NAME="E1:66_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:()"><i><font color="Red">::  thesis: </font></i></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/><b>then reconsider </b><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="66_1_4_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_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="c12">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="c12">CC</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_4_2_1_1"><b>suppose </b></a><a NAME="E1:66_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="c12">CC</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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"><b>hence </b><a NAME="E2:66_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>
 <b>by </b><i><a class="txt" href="#E7:66_1_4"><i><font color="Green" title="E58">A40</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_4_2_1_2"><b>suppose </b></a><a NAME="E1:66_1_4_2_1_2"/><i><font color="Green" title="E60">A42</font></i>: 
<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="c12">CC</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="E2:66_1_4_2_1_2"/><b>then </b>
( not <font color="Maroon" title="c12">CC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c12">CC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> )
 <b>by </b><i><a class="ref" href="pencil_1.html#T12" target="_self" title="PENCIL_1:th.12">Th12</a></i>;<br/><b>then consider </b><font color="Maroon" title="c21">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:66_1_4_2_1_2"/><i><font color="Green" title="E61">A43</font></i>: 
<font color="Maroon" title="c12">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>
 <b>by </b><i><a class="ref" href="realset1.html#D4" title="REALSET1:def.4">REALSET1:def 4</a></i>;<br/><a NAME="E5:66_1_4_2_1_2"/><i><font color="Green" title="E62">A44</font></i>: 
<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="c12">CC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font>
 <b>by </b><i><a class="txt" href="#E25:66_1"><i><font color="Green" title="E55">A19</font></i></a>, <a class="txt" href="#E7:66_1_4"><i><font color="Green" title="E58">A40</font></i></a>, <a class="txt" href="#E1:66_1_4_2_1_2"><i><font color="Green" title="E60">A42</font></i></a></i>;<br/><a NAME="E6:66_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>
 <b>by </b><i><a class="txt" href="#E4:66_1_4"><i><font color="Green" title="E57">A39</font></i></a></i>;<br/><a NAME="E7:66_1_4_2_1_2"/><b>then </b>
<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>
 <b>by </b><i><a class="txt" href="#E1:66_1_4_2_1_2"><i><font color="Green" title="E60">A42</font></i></a>, <a class="txt" href="#E4:66_1_4_2_1_2"><i><font color="Green" title="E61">A43</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><b>hence </b><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>
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
<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>
<b>by </b><i><a class="txt" href="#E4:66_1_4_2_1_2"><i><font color="Green" title="E61">A43</font></i></a>, <a class="txt" href="#E5:66_1_4_2_1_2"><i><font color="Green" title="E62">A44</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E11:66_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>
 <b>by </b><i><a class="txt" href="#E4:66_1_4"><i><font color="Green" title="E57">A39</font></i></a>, <a class="txt" href="#E7:66_1_4"><i><font color="Green" title="E58">A40</font></i></a>, <a class="txt" href="#E9:66_1_4"><i><font color="Green" title="E59">A41</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>let </b><font color="Maroon" title="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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>assume </b><a NAME="E27:66_1"/><i><font color="Green" title="E57">A45</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>thus </b><a NAME="E28:66_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:()"><i><font color="Red">::  thesis: </font></i></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="66_1_5"><b>proof </b></a><div class="add">

<b>let </b><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>; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>assume </b><a NAME="E1:66_1_5"/><i><font color="Green" title="E58">A46</font></i>: 
( <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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>thus </b><a NAME="E2:66_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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_5_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_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="66_1_5_1_1_1"><b>suppose </b></a><a NAME="E1:66_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:()"><i><font color="Red">::  thesis: </font></i></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"><b>hence </b><a NAME="E2:66_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> 
 <b>by </b><i><a class="ref" href="pencil_1.html#D1" target="_self" title="PENCIL_1:def.1">Def1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_5_1_1_2"><b>suppose </b></a><a NAME="E1:66_1_5_1_1_2"/><i><font color="Green" title="E59">A47</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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"><b>reconsider </b><font color="Maroon" title="c18">Q</font> = <font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<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#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> <b>by </b><i><a class="ref" href="pencil_1.html#T25" target="_self" title="PENCIL_1:th.25">Th25</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">R</font> = <font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<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#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> <b>by </b><i><a class="ref" href="pencil_1.html#T25" target="_self" title="PENCIL_1:th.25">Th25</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">Q1</font> = <font color="Maroon" title="c18">Q</font>, <font color="Maroon" title="c21">R1</font> = <font color="Maroon" title="c19">R</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> ;<br/><i><font color="Green" title="E60">A48</font></i>:  <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>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">B</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/><a NAME="E6:66_1_5_1_1_2"/>
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="66_1_5_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>assume </b><a NAME="E1:66_1_5_1_1_2_2"/><i><font color="Green" title="E61">A49</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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="E2:66_1_5_1_1_2_2"/><b>then </b>
<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>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:66_1_5_1_1_2_2"/><b>then </b><i><font color="Green" title="E62">A50</font></i>: 
<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="c8">a1</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_5_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_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="66_1_5_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:66_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:()"><i><font color="Red">::  thesis: </font></i></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="E2:66_1_5_1_1_2_2_1_1"/><b>then </b>
<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="c8">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E3:66_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>
 <b>by </b><i><a class="txt" href="#E4:66_1"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="#E26:66_1"><i><font color="Green" title="E56">A35</font></i></a>, <a class="txt" href="#E1:66_1_5_1_1_2_2"><i><font color="Green" title="E61">A49</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_5_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:66_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:()"><i><font color="Red">::  thesis: </font></i></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"><b>hence </b><a NAME="E2:66_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>
 <b>by </b><i><a class="txt" href="#E27:66_1"><i><font color="Green" title="E57">A45</font></i></a>, <a class="txt" href="#E1:66_1_5"><i><font color="Green" title="E58">A46</font></i></a>, <a class="txt" href="#E3:66_1_5_1_1_2_2"><i><font color="Green" title="E62">A50</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E7:66_1_5_1_1_2"/><b>then </b><i><font color="Green" title="E61">A51</font></i>: 
<font color="Maroon" title="c18">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>
 <b>by </b><i><a class="txt" href="#E5:66_1_5_1_1_2"><i><font color="Green" title="E60">A48</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><i><font color="Green" title="E62">A52</font></i>:  <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>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">B</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/><a NAME="E9:66_1_5_1_1_2"/>
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="66_1_5_1_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>assume </b><a NAME="E1:66_1_5_1_1_2_4"/><i><font color="Green" title="E63">A53</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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="E2:66_1_5_1_1_2_4"/><b>then </b>
<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>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:66_1_5_1_1_2_4"/><b>then </b><i><font color="Green" title="E64">A54</font></i>: 
<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="c8">a1</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_5_1_1_2_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_5_1_1_2_4_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="66_1_5_1_1_2_4_1_1"><b>suppose </b></a><a NAME="E1:66_1_5_1_1_2_4_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:()"><i><font color="Red">::  thesis: </font></i></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="E2:66_1_5_1_1_2_4_1_1"/><b>then </b>
<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="c8">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E3:66_1_5_1_1_2_4_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>
 <b>by </b><i><a class="txt" href="#E4:66_1"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="#E26:66_1"><i><font color="Green" title="E56">A35</font></i></a>, <a class="txt" href="#E1:66_1_5_1_1_2_4"><i><font color="Green" title="E63">A53</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_5_1_1_2_4_1_2"><b>suppose </b></a><a NAME="E1:66_1_5_1_1_2_4_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:()"><i><font color="Red">::  thesis: </font></i></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"><b>hence </b><a NAME="E2:66_1_5_1_1_2_4_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>
 <b>by </b><i><a class="txt" href="#E27:66_1"><i><font color="Green" title="E57">A45</font></i></a>, <a class="txt" href="#E1:66_1_5"><i><font color="Green" title="E58">A46</font></i></a>, <a class="txt" href="#E3:66_1_5_1_1_2_4"><i><font color="Green" title="E64">A54</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E10:66_1_5_1_1_2"/><b>then </b>
<font color="Maroon" title="c19">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>
 <b>by </b><i><a class="txt" href="#E8:66_1_5_1_1_2"><i><font color="Green" title="E62">A52</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><a NAME="E11:66_1_5_1_1_2"/><b>then </b><i><font color="Green" title="E63">A55</font></i>: 
<font color="Maroon" title="c18">Q</font>,<font color="Maroon" title="c19">R</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a> 
 <b>by </b><i><a class="txt" href="#E1:66_1"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="#E26:66_1"><i><font color="Green" title="E56">A35</font></i></a>, <a class="txt" href="#E7:66_1_5_1_1_2"><i><font color="Green" title="E61">A51</font></i></a>, <a class="ref" href="pencil_1.html#D3" target="_self" title="PENCIL_1:def.3">Def3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_5_1_1_2_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_1_5_1_1_2_5"/><i><font color="Green" title="E64">A56</font></i>: 
<font color="Maroon" title="c18">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:66_1_5_1_1_2_5"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:66_1_5_1_1_2_5"/><b>then </b><i><font color="Green" title="E65">A57</font></i>: 
<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>
 <b>by </b><i><a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><a NAME="E4:66_1_5_1_1_2_5"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><b>hence </b><a NAME="E5:66_1_5_1_1_2_5"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:66_1_5_1_1_2"><i><font color="Green" title="E59">A47</font></i></a>, <a class="txt" href="#E1:66_1_5_1_1_2_5"><i><font color="Green" title="E64">A56</font></i></a>, <a class="txt" href="#E3:66_1_5_1_1_2_5"><i><font color="Green" title="E65">A57</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>then consider </b><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#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span><b> such that </b><br/><a NAME="E14:66_1_5_1_1_2"/><i><font color="Green" title="E64">A58</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">Q</font>,<font color="Maroon" title="c19">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>
 <b>by </b><i><a class="txt" href="#E11:66_1_5_1_1_2"><i><font color="Green" title="E63">A55</font></i></a>, <a class="ref" href="pencil_1.html#D1" target="_self" title="PENCIL_1:def.1">Def1</a></i>;<br/><b>consider </b><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#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E16:66_1_5_1_1_2"/><i><font color="Green" title="E65">A59</font></i>: 
( <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> &amp; <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> )
 <b>by </b><i><a class="ref" href="pencil_1.html#T24" target="_self" title="PENCIL_1:th.24">Th24</a></i>;<br/><a NAME="E17:66_1_5_1_1_2"/><i><font color="Green" title="E66">A60</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="#E14:66_1_5_1_1_2"><i><font color="Green" title="E64">A58</font></i></a>, <a class="txt" href="#E16:66_1_5_1_1_2"><i><font color="Green" title="E65">A59</font></i></a>, <a class="ref" href="zfmisc_1.html#T38" title="ZFMISC_1:th.38">ZFMISC_1:38</a></i>;<br/><a NAME="E18:66_1_5_1_1_2"/>
 <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>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E19:66_1_5_1_1_2"/><b>then </b><i><font color="Green" title="E67">A61</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="#E17:66_1_5_1_1_2"><i><font color="Green" title="E66">A60</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><a NAME="E20:66_1_5_1_1_2"/><i><font color="Green" title="E68">A62</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="#E14:66_1_5_1_1_2"><i><font color="Green" title="E64">A58</font></i></a>, <a class="txt" href="#E16:66_1_5_1_1_2"><i><font color="Green" title="E65">A59</font></i></a>, <a class="ref" href="zfmisc_1.html#T38" title="ZFMISC_1:th.38">ZFMISC_1:38</a></i>;<br/><a NAME="E21:66_1_5_1_1_2"/>
 <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>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E22:66_1_5_1_1_2"/><b>then </b><i><font color="Green" title="E69">A63</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="#E20:66_1_5_1_1_2"><i><font color="Green" title="E68">A62</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_5_1_1_2_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_1_5_1_1_2_6"/><i><font color="Green" title="E70">A64</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:66_1_5_1_1_2_6"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:66_1_5_1_1_2_6"/><b>then </b><i><font color="Green" title="E71">A65</font></i>: 
<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>
 <b>by </b><i><a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><a NAME="E4:66_1_5_1_1_2_6"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><b>hence </b><a NAME="E5:66_1_5_1_1_2_6"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:66_1_5_1_1_2"><i><font color="Green" title="E59">A47</font></i></a>, <a class="txt" href="#E1:66_1_5_1_1_2_6"><i><font color="Green" title="E70">A64</font></i></a>, <a class="txt" href="#E3:66_1_5_1_1_2_6"><i><font color="Green" title="E71">A65</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E24:66_1_5_1_1_2"/><b>then </b>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><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>
 <b>by </b><i><a class="txt" href="#E19:66_1_5_1_1_2"><i><font color="Green" title="E67">A61</font></i></a>, <a class="txt" href="#E22:66_1_5_1_1_2"><i><font color="Green" title="E69">A63</font></i></a>, <a class="ref" href="pencil_1.html#T2" target="_self" title="PENCIL_1:th.2">Th2</a></i>;<br/><a NAME="E25:66_1_5_1_1_2"/><b>then </b><i><font color="Green" title="E70">A66</font></i>: 
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>
 <b>by </b><i><a class="ref" href="pencil_1.html#T4" target="_self" title="PENCIL_1:th.4">Th4</a></i>;<br/><a NAME="E26:66_1_5_1_1_2"/><b>then </b><i><font color="Green" title="E71">A67</font></i>: 
 <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>
 <b>by </b><i><a class="ref" href="pencil_1.html#D21" target="_self" title="PENCIL_1:def.21">Def21</a></i>;<br/><b>reconsider </b><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> <b>by </b><i><a class="txt" href="#E16:66_1_5_1_1_2"><i><font color="Green" title="E65">A59</font></i></a>, <a class="txt" href="#E25:66_1_5_1_1_2"><i><font color="Green" title="E70">A66</font></i></a>, <a class="ref" href="pencil_1.html#D21" target="_self" title="PENCIL_1:def.21">Def21</a></i>;<br/><a NAME="E28:66_1_5_1_1_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E29:66_1_5_1_1_2"/><b>then </b><i><font color="Green" title="E72">A68</font></i>: 
<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>
 <b>by </b><i><a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><a NAME="E30:66_1_5_1_1_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E31:66_1_5_1_1_2"/><b>then </b>
<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>
 <b>by </b><i><a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><a NAME="E32:66_1_5_1_1_2"/><b>then </b>
<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>
 <b>by </b><i><a class="txt" href="#E19:66_1_5_1_1_2"><i><font color="Green" title="E67">A61</font></i></a>, <a class="txt" href="#E22:66_1_5_1_1_2"><i><font color="Green" title="E69">A63</font></i></a>, <a class="txt" href="#E26:66_1_5_1_1_2"><i><font color="Green" title="E71">A67</font></i></a>, <a class="txt" href="#E29:66_1_5_1_1_2"><i><font color="Green" title="E72">A68</font></i></a>, <a class="ref" href="zfmisc_1.html#T38" title="ZFMISC_1:th.38">ZFMISC_1:38</a></i>;<br/><b>hence </b><a NAME="E33:66_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> 
 <b>by </b><i><a class="ref" href="pencil_1.html#D1" target="_self" title="PENCIL_1:def.1">Def1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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


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

<b>thus </b><a NAME="E29:66_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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_6"><b>proof </b></a><div class="add">

<b>let </b><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>; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></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="E1:66_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="E2:66_1_6"/><b>then </b><i><font color="Green" title="E58">A69</font></i>: 
<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/>
<b>assume </b><a NAME="E3:66_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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>then consider </b><font color="Maroon" title="c17">x</font>, <font color="Maroon" title="c18">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:66_1_6"/><i><font color="Green" title="E59">A70</font></i>: 
( <font color="Maroon" title="c17">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> &amp; <font color="Maroon" title="c18">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> &amp; <font color="Maroon" title="c17">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">y</font> )
 <b>by </b><i><a class="ref" href="pencil_1.html#T2" target="_self" title="PENCIL_1:th.2">Th2</a></i>;<br/>
<a NAME="E6:66_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>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D6" target="_self" title="PENCIL_1:def.6">Def6</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c19">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>  <b>by </b><i><a class="ref" href="pencil_1.html#T4" target="_self" title="PENCIL_1:th.4">Th4</a></i>;<br/>
<b>set </b><font color="Maroon" title="c20">x1</font> = <font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c17">x</font>;<br/>
<b>set </b><font color="Maroon" title="c21">y1</font> = <font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c18">y</font>;<br/>
<a NAME="E8:66_1_6"/><i><font color="Green" title="E60">A71</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c17">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="c17">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c18">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="c18">y</font> )
 
<b>by </b><i><a class="txt" href="#E7:66_1"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/>
<a NAME="E9:66_1_6"/>
<font color="Maroon" title="c14">B</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c19">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="66_1_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c22">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c22">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="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c19">L1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">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="c22">o</font> )</span><br/>

<b>assume </b><a NAME="E1:66_1_6_1"/><i><font color="Green" title="E61">A72</font></i>: 
<font color="Maroon" title="c22">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c19">L1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">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="c22">o</font></span><br/>

<b>thus </b><a NAME="E2:66_1_6_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c19">L1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">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="c22">o</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_6_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_6_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_6_1_1_1"/>
( <font color="Maroon" title="c22">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="c22">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="66_1_6_1_1_1_1"><b>suppose </b></a><a NAME="E1:66_1_6_1_1_1_1"/>
<font color="Maroon" title="c22">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c19">L1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">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="c22">o</font></span><br/><div class="add"><a NAME="E2:66_1_6_1_1_1_1"/><b>then </b><i><font color="Green" title="E62">A73</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c19">L1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">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="c22">o</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><a NAME="E3:66_1_6_1_1_1_1"/>
<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>
 <b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/><b>hence </b><a NAME="E4:66_1_6_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c19">L1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">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="c22">o</font>
 <b>by </b><i><a class="txt" href="#E1:66_1_6_1"><i><font color="Green" title="E61">A72</font></i></a>, <a class="txt" href="#E2:66_1_6_1_1_1_1"><i><font color="Green" title="E62">A73</font></i></a>, <a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_6_1_1_1_2"><b>suppose </b></a><a NAME="E1:66_1_6_1_1_1_2"/><i><font color="Green" title="E62">A74</font></i>: 
<font color="Maroon" title="c22">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c19">L1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">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="c22">o</font></span><br/><div class="add"><a NAME="E2:66_1_6_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c19">L1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">L</font>
 <b>by </b><i><a class="txt" href="#E24:66_1"><i><font color="Green" title="E54">A18</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><b>hence </b><a NAME="E3:66_1_6_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c19">L1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">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="c22">o</font>
 <b>by </b><i><a class="txt" href="#E2:66_1_6"><i><font color="Green" title="E58">A69</font></i></a>, <a class="txt" href="#E1:66_1_6_1_1_1_2"><i><font color="Green" title="E62">A74</font></i></a>, <a class="ref" href="yellow_6.html#T9" title="YELLOW_6:th.9">YELLOW_6:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c22">L2</font> = <font color="Maroon" title="c14">B</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c19">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#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="ref" href="pencil_1.html#T27" target="_self" title="PENCIL_1:th.27">Th27</a>, <a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E11:66_1_6"/><i><font color="Green" title="E61">A75</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c22">L2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E12:66_1_6"/><i><font color="Green" title="E62">A76</font></i>: 
<font color="Maroon" title="c22">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="c19">L1</font>
 
<b>by </b><i><a class="txt" href="#E24:66_1"><i><font color="Green" title="E54">A18</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/>
<a NAME="E13:66_1_6"/><b>then </b><i><font color="Green" title="E63">A77</font></i>: 
 <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="c22">L2</font>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D21" target="_self" title="PENCIL_1:def.21">Def21</a></i>;<br/>
<a NAME="E14:66_1_6"/>
 <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>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E15:66_1_6"/><b>then </b>
<font color="Maroon" title="c22">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="c22">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="c22">L2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E13:66_1_6"><i><font color="Green" title="E63">A77</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c23">L3</font> =  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c22">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#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> <b>by </b><i><a class="ref" href="pencil_1.html#T24" target="_self" title="PENCIL_1:th.24">Th24</a></i>;<br/>
<a NAME="E17:66_1_6"/><i><font color="Green" title="E64">A78</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c17">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_6_2"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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="c22">L2</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c17">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="c22">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><b>assume </b><a NAME="E1:66_1_6_2"/><i><font color="Green" title="E65">A79</font></i>: 
<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="c22">L2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c17">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="c22">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="66_1_6_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_6_2_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="66_1_6_2_1_1"><b>suppose </b></a><a NAME="E1:66_1_6_2_1_1"/><i><font color="Green" title="E66">A80</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c17">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="c22">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="E2:66_1_6_2_1_1"/><b>then </b><i><font color="Green" title="E67">A81</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c17">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="c8">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">o</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><a NAME="E3:66_1_6_2_1_1"/>
<font color="Maroon" title="c8">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>
 <b>by </b><i><a class="txt" href="#E4:66_1"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="#E24:66_1"><i><font color="Green" title="E54">A18</font></i></a>, <a class="txt" href="#E26:66_1"><i><font color="Green" title="E56">A35</font></i></a>, <a class="txt" href="#E11:66_1_6"><i><font color="Green" title="E61">A75</font></i></a>, <a class="txt" href="#E1:66_1_6_2"><i><font color="Green" title="E65">A79</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><b>hence </b><a NAME="E4:66_1_6_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c17">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="c22">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">o</font>
 <b>by </b><i><a class="txt" href="#E1:66_1_6_2_1_1"><i><font color="Green" title="E66">A80</font></i></a>, <a class="txt" href="#E2:66_1_6_2_1_1"><i><font color="Green" title="E67">A81</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_6_2_1_2"><b>suppose </b></a><a NAME="E1:66_1_6_2_1_2"/><i><font color="Green" title="E66">A82</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c17">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="c22">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="E2:66_1_6_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c17">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="c17">x</font>
 <b>by </b><i><a class="txt" href="#E7:66_1"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><b>hence </b><a NAME="E3:66_1_6_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c17">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="c22">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">o</font>
 <b>by </b><i><a class="txt" href="#E5:66_1_6"><i><font color="Green" title="E59">A70</font></i></a>, <a class="txt" href="#E12:66_1_6"><i><font color="Green" title="E62">A76</font></i></a>, <a class="txt" href="#E1:66_1_6_2_1_2"><i><font color="Green" title="E66">A82</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E19:66_1_6"/><b>then </b><i><font color="Green" title="E65">A83</font></i>: 
<font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">L3</font>
 
<b>by </b><i><a class="txt" href="#E11:66_1_6"><i><font color="Green" title="E61">A75</font></i></a>, <a class="txt" href="#E17:66_1_6"><i><font color="Green" title="E64">A78</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_6_3"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c17">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/><b>assume </b><a NAME="E1:66_1_6_3"/><i><font color="Green" title="E66">A84</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c17">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="66_1_6_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_6_3_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="66_1_6_3_1_1"><b>suppose </b></a><a NAME="E1:66_1_6_3_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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c17">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="E2:66_1_6_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c17">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="c8">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">o</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E3:66_1_6_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c17">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>
 <b>by </b><i><a class="txt" href="#E4:66_1"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="#E24:66_1"><i><font color="Green" title="E54">A18</font></i></a>, <a class="txt" href="#E26:66_1"><i><font color="Green" title="E56">A35</font></i></a>, <a class="txt" href="#E1:66_1_6_3"><i><font color="Green" title="E66">A84</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_6_3_1_2"><b>suppose </b></a><a NAME="E1:66_1_6_3_1_2"/><i><font color="Green" title="E67">A85</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c17">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="E2:66_1_6_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c17">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="c17">x</font>
 <b>by </b><i><a class="txt" href="#E7:66_1"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><b>hence </b><a NAME="E3:66_1_6_3_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c17">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>
 <b>by </b><i><a class="txt" href="#E27:66_1"><i><font color="Green" title="E57">A45</font></i></a>, <a class="txt" href="#E5:66_1_6"><i><font color="Green" title="E59">A70</font></i></a>, <a class="txt" href="#E1:66_1_6_3_1_2"><i><font color="Green" title="E67">A85</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E21:66_1_6"/><b>then </b>
<font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font>
 
<b>by </b><i><a class="txt" href="#E24:66_1"><i><font color="Green" title="E54">A18</font></i></a>, <a class="txt" href="#E26:66_1"><i><font color="Green" title="E56">A35</font></i></a>, <a class="txt" href="#E17:66_1_6"><i><font color="Green" title="E64">A78</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E22:66_1_6"/><b>then </b><i><font color="Green" title="E66">A86</font></i>: 
<font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">L3</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">S</font>
 
<b>by </b><i><a class="txt" href="#E19:66_1_6"><i><font color="Green" title="E65">A83</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E23:66_1_6"/><i><font color="Green" title="E67">A87</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c18">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_6_4"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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="c22">L2</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c18">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="c22">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><b>assume </b><a NAME="E1:66_1_6_4"/><i><font color="Green" title="E68">A88</font></i>: 
<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="c22">L2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c18">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="c22">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="66_1_6_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_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="66_1_6_4_1_1"><b>suppose </b></a><a NAME="E1:66_1_6_4_1_1"/><i><font color="Green" title="E69">A89</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c18">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="c22">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="E2:66_1_6_4_1_1"/><b>then </b><i><font color="Green" title="E70">A90</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c18">y</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="c8">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">o</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><a NAME="E3:66_1_6_4_1_1"/>
<font color="Maroon" title="c8">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>
 <b>by </b><i><a class="txt" href="#E4:66_1"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="#E24:66_1"><i><font color="Green" title="E54">A18</font></i></a>, <a class="txt" href="#E26:66_1"><i><font color="Green" title="E56">A35</font></i></a>, <a class="txt" href="#E11:66_1_6"><i><font color="Green" title="E61">A75</font></i></a>, <a class="txt" href="#E1:66_1_6_4"><i><font color="Green" title="E68">A88</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><b>hence </b><a NAME="E4:66_1_6_4_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c18">y</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="c22">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">o</font>
 <b>by </b><i><a class="txt" href="#E1:66_1_6_4_1_1"><i><font color="Green" title="E69">A89</font></i></a>, <a class="txt" href="#E2:66_1_6_4_1_1"><i><font color="Green" title="E70">A90</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_6_4_1_2"><b>suppose </b></a><a NAME="E1:66_1_6_4_1_2"/><i><font color="Green" title="E69">A91</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c18">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="c22">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="E2:66_1_6_4_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c18">y</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="c18">y</font>
 <b>by </b><i><a class="txt" href="#E7:66_1"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><b>hence </b><a NAME="E3:66_1_6_4_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c18">y</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="c22">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">o</font>
 <b>by </b><i><a class="txt" href="#E5:66_1_6"><i><font color="Green" title="E59">A70</font></i></a>, <a class="txt" href="#E12:66_1_6"><i><font color="Green" title="E62">A76</font></i></a>, <a class="txt" href="#E1:66_1_6_4_1_2"><i><font color="Green" title="E69">A91</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E25:66_1_6"/><b>then </b><i><font color="Green" title="E68">A92</font></i>: 
<font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">L3</font>
 
<b>by </b><i><a class="txt" href="#E11:66_1_6"><i><font color="Green" title="E61">A75</font></i></a>, <a class="txt" href="#E23:66_1_6"><i><font color="Green" title="E67">A87</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_6_5"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c18">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/><b>assume </b><a NAME="E1:66_1_6_5"/><i><font color="Green" title="E69">A93</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c18">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="66_1_6_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_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="66_1_6_5_1_1"><b>suppose </b></a><a NAME="E1:66_1_6_5_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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c18">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="E2:66_1_6_5_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c18">y</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="c8">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">o</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E3:66_1_6_5_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c18">y</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>
 <b>by </b><i><a class="txt" href="#E4:66_1"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="#E24:66_1"><i><font color="Green" title="E54">A18</font></i></a>, <a class="txt" href="#E26:66_1"><i><font color="Green" title="E56">A35</font></i></a>, <a class="txt" href="#E1:66_1_6_5"><i><font color="Green" title="E69">A93</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_6_5_1_2"><b>suppose </b></a><a NAME="E1:66_1_6_5_1_2"/><i><font color="Green" title="E70">A94</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c18">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="E2:66_1_6_5_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c18">y</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="c18">y</font>
 <b>by </b><i><a class="txt" href="#E7:66_1"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><b>hence </b><a NAME="E3:66_1_6_5_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c18">y</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>
 <b>by </b><i><a class="txt" href="#E27:66_1"><i><font color="Green" title="E57">A45</font></i></a>, <a class="txt" href="#E5:66_1_6"><i><font color="Green" title="E59">A70</font></i></a>, <a class="txt" href="#E1:66_1_6_5_1_2"><i><font color="Green" title="E70">A94</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E27:66_1_6"/><b>then </b>
<font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font>
 
<b>by </b><i><a class="txt" href="#E24:66_1"><i><font color="Green" title="E54">A18</font></i></a>, <a class="txt" href="#E26:66_1"><i><font color="Green" title="E56">A35</font></i></a>, <a class="txt" href="#E23:66_1_6"><i><font color="Green" title="E67">A87</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E28:66_1_6"/><b>then </b>
<font color="Maroon" title="c8">a1</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">L3</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">S</font>
 
<b>by </b><i><a class="txt" href="#E25:66_1_6"><i><font color="Green" title="E68">A92</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E29:66_1_6"/><b>then </b>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">L3</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">S</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E5:66_1_6"><i><font color="Green" title="E59">A70</font></i></a>, <a class="txt" href="#E8:66_1_6"><i><font color="Green" title="E60">A71</font></i></a>, <a class="txt" href="#E22:66_1_6"><i><font color="Green" title="E66">A86</font></i></a>, <a class="ref" href="pencil_1.html#T2" target="_self" title="PENCIL_1:th.2">Th2</a></i>;<br/>
<a NAME="E30:66_1_6"/><b>then </b><i><font color="Green" title="E69">A95</font></i>: 
<font color="Maroon" title="c23">L3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">S</font>
 
<b>by </b><i><a class="txt" href="#E1:66_1"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="pencil_1.html#D2" target="_self" title="PENCIL_1:def.2">Def2</a></i>;<br/>
<b>thus </b><a NAME="E31:66_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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_6_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c24">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="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/>

<b>assume </b><a NAME="E1:66_1_6_6"/><i><font color="Green" title="E70">A96</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>consider </b><font color="Maroon" title="c25">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:66_1_6_6"/><i><font color="Green" title="E71">A97</font></i>: 
<font color="Maroon" title="c25">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">L3</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c26">F</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:66_1_6_6"/><i><font color="Green" title="E72">A98</font></i>: 
( <font color="Maroon" title="c26">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="Maroon" title="c26">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="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> <font color="Maroon" title="c1">I</font> holds <br/><font color="Maroon" title="c26">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">o</font> ) )
 <b>by </b><i><a class="txt" href="#E11:66_1_6"><i><font color="Green" title="E61">A75</font></i></a>, <a class="txt" href="#E3:66_1_6_6"><i><font color="Green" title="E71">A97</font></i></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c27">f</font> = <font color="Maroon" title="c25">f</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="txt" href="#E5:66_1_6_6"><i><font color="Green" title="E72">A98</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E7:66_1_6_6"/><i><font color="Green" title="E73">A99</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">f</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<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="c22">L2</font>
 
<b>by </b><i><a class="txt" href="#E11:66_1_6"><i><font color="Green" title="E61">A75</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_6_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c28">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c28">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="c22">L2</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c27">f</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<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="c22">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><b>assume </b><a NAME="E1:66_1_6_6_1"/><i><font color="Green" title="E74">A100</font></i>: 
<font color="Maroon" title="c28">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="c22">L2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c27">f</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<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="c22">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="66_1_6_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_6_6_1_1"/>
( <font color="Maroon" title="c28">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="c28">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="66_1_6_6_1_1_1"><b>suppose </b></a><a NAME="E1:66_1_6_6_1_1_1"/>
<font color="Maroon" title="c28">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c27">f</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<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="c22">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="E2:66_1_6_6_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c27">f</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<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="c28">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">o</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E3:66_1_6_6_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c27">f</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<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="c28">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">o</font>
 <b>by </b><i><a class="txt" href="#E11:66_1_6"><i><font color="Green" title="E61">A75</font></i></a>, <a class="txt" href="#E5:66_1_6_6"><i><font color="Green" title="E72">A98</font></i></a>, <a class="txt" href="#E1:66_1_6_6_1"><i><font color="Green" title="E74">A100</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_6_6_1_1_2"><b>suppose </b></a><a NAME="E1:66_1_6_6_1_1_2"/><i><font color="Green" title="E75">A101</font></i>: 
<font color="Maroon" title="c28">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c27">f</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<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="c22">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="E2:66_1_6_6_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c27">f</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<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="c28">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">e</font>
 <b>by </b><i><a class="txt" href="#E5:66_1_6_6"><i><font color="Green" title="E72">A98</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><b>hence </b><a NAME="E3:66_1_6_6_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c27">f</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<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="c28">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">o</font>
 <b>by </b><i><a class="txt" href="#E24:66_1"><i><font color="Green" title="E54">A18</font></i></a>, <a class="txt" href="#E1:66_1_6_6"><i><font color="Green" title="E70">A96</font></i></a>, <a class="txt" href="#E1:66_1_6_6_1_1_2"><i><font color="Green" title="E75">A101</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E9:66_1_6_6"/><b>then </b>
<font color="Maroon" title="c27">f</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c24">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">L3</font>
 
<b>by </b><i><a class="txt" href="#E7:66_1_6_6"><i><font color="Green" title="E73">A99</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E10:66_1_6_6"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c27">f</font> <a href="polynom1.html#K1" title="POLYNOM1: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>,<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>
 
<b>by </b><i><a class="txt" href="#E24:66_1"><i><font color="Green" title="E54">A18</font></i></a>, <a class="txt" href="#E26:66_1"><i><font color="Green" title="E56">A35</font></i></a>, <a class="txt" href="#E30:66_1_6"><i><font color="Green" title="E69">A95</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<b>hence </b><a NAME="E11:66_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>
 <b>by </b><i><a class="txt" href="#E27:66_1"><i><font color="Green" title="E57">A45</font></i></a>, <a class="txt" href="#E5:66_1_6_6"><i><font color="Green" title="E72">A98</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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


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


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

<b>given </b><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#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font><b> such that </b><a NAME="E4:66"/><i><font color="Green" title="E45">A102</font></i>: 
( <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> &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="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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>thus </b><a NAME="E5:66"/>
not <font color="Maroon" title="c3">S</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>
 <b>by </b><i><a class="txt" href="#E4:66"><i><font color="Green" title="E45">A102</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="E6:66"/><i><font color="Green" title="E46">A103</font></i>: 
 <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>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>thus </b><a NAME="E7:66"/>
<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:()"><i><font color="Red">::  thesis: </font></i></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="66_2"><b>proof </b></a><div class="add">

<b>let </b><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#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>assume </b><a NAME="E1:66_2"/><i><font color="Green" title="E47">A104</font></i>: 
( <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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="66_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_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="66_2_1_1"><b>suppose </b></a><a NAME="E1:66_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:()"><i><font color="Red">::  thesis: </font></i></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"><b>hence </b><a NAME="E2:66_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> 
 <b>by </b><i><a class="ref" href="pencil_1.html#D1" target="_self" title="PENCIL_1:def.1">Def1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_2"><b>suppose </b></a><a NAME="E1:66_2_1_2"/><i><font color="Green" title="E48">A105</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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"><b>consider </b><font color="Maroon" title="c7">x1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:66_2_1_2"/><i><font color="Green" title="E49">A106</font></i>: 
( <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">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> &amp; ( 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">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> ) )
 <b>by </b><i><a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">x1</font> = <font color="Maroon" title="c7">x1</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="txt" href="#E3:66_2_1_2"><i><font color="Green" title="E49">A106</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">y1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E6:66_2_1_2"/><i><font color="Green" title="E50">A107</font></i>: 
( <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">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> &amp; ( 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">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> ) )
 <b>by </b><i><a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">y1</font> = <font color="Maroon" title="c9">y1</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="txt" href="#E6:66_2_1_2"><i><font color="Green" title="E50">A107</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><div><i><font color="Green" title="E51">A108</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_2_1_2_1"/><i><font color="Green" title="E52">A109</font></i>: 
<font color="Maroon" title="c8">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="c10">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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="c8">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="c10">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><b>assume </b><a NAME="E1:66_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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">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="c10">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="66_2_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_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="66_2_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:66_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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">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="c10">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"><b>hence </b><a NAME="E2:66_2_1_2_1_1_1_1"/>
<font color="Maroon" title="c8">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="c10">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font>
 <b>by </b><i><a class="txt" href="#E4:66"><i><font color="Green" title="E45">A102</font></i></a>, <a class="txt" href="#E1:66_2"><i><font color="Green" title="E47">A104</font></i></a>, <a class="txt" href="#E3:66_2_1_2"><i><font color="Green" title="E49">A106</font></i></a>, <a class="txt" href="#E6:66_2_1_2"><i><font color="Green" title="E50">A107</font></i></a>, <a class="ref" href="pencil_1.html#T23" target="_self" title="PENCIL_1:th.23">Th23</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:66_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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">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="c10">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"><b>hence </b><a NAME="E2:66_2_1_2_1_1_1_2"/>
<font color="Maroon" title="c8">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="c10">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font>
 <b>by </b><i><a class="txt" href="#E1:66_2_1_2_1"><i><font color="Green" title="E52">A109</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:66_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:66_2_1_2"><i><font color="Green" title="E48">A105</font></i></a>, <a class="txt" href="#E3:66_2_1_2"><i><font color="Green" title="E49">A106</font></i></a>, <a class="txt" href="#E6:66_2_1_2"><i><font color="Green" title="E50">A107</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E9:66_2_1_2"/>
<font color="Maroon" title="c8">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>
 <b>by </b><i><a class="txt" href="#E1:66"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="#E3:66_2_1_2"><i><font color="Green" title="E49">A106</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">x1i</font> = <font color="Maroon" title="c8">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> <b>by </b><i><a class="ref" href="yellow_6.html#T9" title="YELLOW_6:th.9">YELLOW_6:9</a></i>;<br/><a NAME="E11:66_2_1_2"/>
<font color="Maroon" title="c10">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>
 <b>by </b><i><a class="txt" href="#E1:66"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="#E6:66_2_1_2"><i><font color="Green" title="E50">A107</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c12">y1i</font> = <font color="Maroon" title="c10">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> <b>by </b><i><a class="ref" href="yellow_6.html#T9" title="YELLOW_6:th.9">YELLOW_6:9</a></i>;<br/><a NAME="E13:66_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>
 <b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/><a NAME="E14:66_2_1_2"/><b>then </b>
<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>
 <b>by </b><i><a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/><b>then reconsider </b><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> <b>by </b><i><a class="ref" href="yellow_6.html#T9" title="YELLOW_6:th.9">YELLOW_6:9</a></i>;<br/><a NAME="E16:66_2_1_2"/><i><font color="Green" title="E52">A110</font></i>: 
<font color="Maroon" title="c11">x1i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">C</font>
 <b>by </b><i><a class="txt" href="#E4:66"><i><font color="Green" title="E45">A102</font></i></a>, <a class="txt" href="#E6:66"><i><font color="Green" title="E46">A103</font></i></a>, <a class="txt" href="#E1:66_2"><i><font color="Green" title="E47">A104</font></i></a>, <a class="txt" href="#E3:66_2_1_2"><i><font color="Green" title="E49">A106</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><a NAME="E17:66_2_1_2"/><i><font color="Green" title="E53">A111</font></i>: 
<font color="Maroon" title="c12">y1i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">C</font>
 <b>by </b><i><a class="txt" href="#E4:66"><i><font color="Green" title="E45">A102</font></i></a>, <a class="txt" href="#E6:66"><i><font color="Green" title="E46">A103</font></i></a>, <a class="txt" href="#E1:66_2"><i><font color="Green" title="E47">A104</font></i></a>, <a class="txt" href="#E6:66_2_1_2"><i><font color="Green" title="E50">A107</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><a NAME="E18:66_2_1_2"/>
<font color="Maroon" title="c13">C</font> is <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a>
 <b>by </b><i><a class="txt" href="#E4:66"><i><font color="Green" title="E45">A102</font></i></a></i>;<br/><a NAME="E19:66_2_1_2"/><b>then </b>
<font color="Maroon" title="c11">x1i</font>,<font color="Maroon" title="c12">y1i</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a> 
 <b>by </b><i><a class="txt" href="#E16:66_2_1_2"><i><font color="Green" title="E52">A110</font></i></a>, <a class="txt" href="#E17:66_2_1_2"><i><font color="Green" title="E53">A111</font></i></a>, <a class="ref" href="pencil_1.html#D3" target="_self" title="PENCIL_1:def.3">Def3</a></i>;<br/><b>then consider </b><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><b> such that </b><br/><a NAME="E21:66_2_1_2"/><i><font color="Green" title="E54">A112</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">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>
 <b>by </b><i><a class="txt" href="#E8:66_2_1_2"><i><font color="Green" title="E51">A108</font></i></a>, <a class="ref" href="pencil_1.html#D1" target="_self" title="PENCIL_1:def.1">Def1</a></i>;<br/><a NAME="E22:66_2_1_2"/><i><font color="Green" title="E55">A113</font></i>: 
 <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="c8">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>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E23:66_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/>( not <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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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 <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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 <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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 ( not <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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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 <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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 <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> ) )</span><br/>

<b>assume </b><a NAME="E1:66_2_1_2_2"/>
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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 <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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 <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> )</span><br/>

<b>then </b><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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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="c8">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>
<b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>
<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="c8">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>
<b>by </b><i><a class="ref" href="pzfmisc1.html#D1" title="PZFMISC1:def.1">PZFMISC1:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E3:66_2_1_2_2"/>
( not <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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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 <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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 <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E24:66_2_1_2"/><b>then </b><i><font color="Green" title="E56">A114</font></i>: 
<span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c14">l</font> is <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>
 <b>by </b><i><a class="ref" href="pencil_1.html#D20" target="_self" title="PENCIL_1:def.20">Def20</a></i>;<br/><a NAME="E25:66_2_1_2"/>
<span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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>,<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="66_2_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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/>

<b>assume </b><a NAME="E1:66_2_1_2_3"/><i><font color="Green" title="E57">A115</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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/>

<b>then reconsider </b><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/>
<b>thus </b><a NAME="E3:66_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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_2_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_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="66_2_1_2_3_1_1_1"><b>suppose </b></a><a NAME="E1:66_2_1_2_3_1_1_1"/><i><font color="Green" title="E58">A116</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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="E2:66_2_1_2_3_1_1_1"/><b>then </b>
<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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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>
 <b>by </b><i><a class="txt" href="#E22:66_2_1_2"><i><font color="Green" title="E55">A113</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><a NAME="E3:66_2_1_2_3_1_1_1"/><b>then </b>
<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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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="E4:66_2_1_2_3_1_1_1"/><b>then </b>
<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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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/><b>hence </b><a NAME="E5:66_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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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>
 <b>by </b><i><a class="txt" href="#E1:66_2_1_2_3_1_1_1"><i><font color="Green" title="E58">A116</font></i></a>, <a class="ref" href="yellow_6.html#T9" title="YELLOW_6:th.9">YELLOW_6:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_2_3_1_1_2"><b>suppose </b></a><a NAME="E1:66_2_1_2_3_1_1_2"/>
<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:()"><i><font color="Red">::  thesis: </font></i></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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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"><b>then </b><i><font color="Green" title="E58">A117</font></i>: <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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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="c8">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>
<b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>
<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="c8">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>
<b>by </b><i><a class="txt" href="#E1:66_2_1_2_3"><i><font color="Green" title="E57">A115</font></i></a>, <a class="ref" href="pzfmisc1.html#D1" title="PZFMISC1:def.1">PZFMISC1:def 1</a></i>
;<br/><a NAME="E3:66_2_1_2_3_1_1_2"/>
<font color="Maroon" title="c8">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>
 <b>by </b><i><a class="txt" href="#E1:66"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="#E3:66_2_1_2"><i><font color="Green" title="E49">A106</font></i></a></i>;<br/><a NAME="E4:66_2_1_2_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">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>
 <b>by </b><i><a class="ref" href="yellow_6.html#T9" title="YELLOW_6:th.9">YELLOW_6:9</a></i>;<br/><a NAME="E5:66_2_1_2_3_1_1_2"/><b>then </b>
<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="c8">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>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><b>hence </b><a NAME="E6:66_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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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>
 <b>by </b><i><a class="txt" href="#E2:66_2_1_2_3_1_1_2"><i><font color="Green" title="E58">A117</font></i></a>, <a class="ref" href="yellow_6.html#T9" title="YELLOW_6:th.9">YELLOW_6:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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


</div><b>end;</b></div><a NAME="E26:66_2_1_2"/><b>then </b><i><font color="Green" title="E57">A118</font></i>: 
<span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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>,<font color="Maroon" title="c14">l</font> is    <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/><a NAME="E27:66_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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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>
 <b>by </b><i><a class="txt" href="#E22:66_2_1_2"><i><font color="Green" title="E55">A113</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><b>then reconsider </b><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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> <b>by </b><i><a class="txt" href="#E24:66_2_1_2"><i><font color="Green" title="E56">A114</font></i></a>, <a class="txt" href="#E26:66_2_1_2"><i><font color="Green" title="E57">A118</font></i></a>, <a class="ref" href="pencil_1.html#D22" target="_self" title="PENCIL_1:def.22">Def22</a></i>;<br/><i><font color="Green" title="E58">A119</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">x1</font> = 
<font color="Maroon" title="c1">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/><a NAME="E30:66_2_1_2"/>
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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font></span>)</span> holds <br/><font color="Maroon" title="c8">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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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="66_2_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c16">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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font></span>)</span> implies <font color="Maroon" title="c8">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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="c16">a</font> )</span><br/>

<b>assume </b><a NAME="E1:66_2_1_2_5"/>
<font color="Maroon" title="c16">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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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="c16">a</font></span><br/>

<a NAME="E2:66_2_1_2_5"/><b>then </b><i><font color="Green" title="E59">A120</font></i>: 
<font color="Maroon" title="c16">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_2_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_2_1_2_5_1"/>
( <font color="Maroon" title="c16">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="c16">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="66_2_1_2_5_1_1"><b>suppose </b></a><a NAME="E1:66_2_1_2_5_1_1"/><i><font color="Green" title="E60">A121</font></i>: 
<font color="Maroon" title="c16">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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="c16">a</font></span><br/><div class="add"><a NAME="E2:66_2_1_2_5_1_1"/><b>then </b>
<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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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="c16">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">l</font>
 <b>by </b><i><a class="txt" href="#E22:66_2_1_2"><i><font color="Green" title="E55">A113</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><b>hence </b><a NAME="E3:66_2_1_2_5_1_1"/>
<font color="Maroon" title="c8">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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="c16">a</font>
 <b>by </b><i><a class="txt" href="#E21:66_2_1_2"><i><font color="Green" title="E54">A112</font></i></a>, <a class="txt" href="#E1:66_2_1_2_5_1_1"><i><font color="Green" title="E60">A121</font></i></a>, <a class="ref" href="zfmisc_1.html#T38" title="ZFMISC_1:th.38">ZFMISC_1:38</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_2_5_1_2"><b>suppose </b></a><a NAME="E1:66_2_1_2_5_1_2"/><i><font color="Green" title="E60">A122</font></i>: 
<font color="Maroon" title="c16">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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="c16">a</font></span><br/><div class="add"><a NAME="E2:66_2_1_2_5_1_2"/>
<font color="Maroon" title="c8">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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="c8">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">a</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:66_2_1_2_5_1_2"/><b>then </b>
<font color="Maroon" title="c8">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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="c8">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="c16">a</font>
 <b>by </b><i><a class="txt" href="#E2:66_2_1_2_5"><i><font color="Green" title="E59">A120</font></i></a>, <a class="ref" href="pzfmisc1.html#D1" title="PZFMISC1:def.1">PZFMISC1:def 1</a></i>;<br/><b>hence </b><a NAME="E4:66_2_1_2_5_1_2"/>
<font color="Maroon" title="c8">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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="c16">a</font>
 <b>by </b><i><a class="txt" href="#E1:66_2_1_2_5_1_2"><i><font color="Green" title="E60">A122</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E31:66_2_1_2"/><b>then </b><i><font color="Green" title="E59">A123</font></i>: 
<font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">L</font>
 <b>by </b><i><a class="txt" href="#E29:66_2_1_2"><i><font color="Green" title="E58">A119</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><i><font color="Green" title="E60">A124</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">y1</font> = 
<font color="Maroon" title="c1">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/><a NAME="E33:66_2_1_2"/>
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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font></span>)</span> holds <br/><font color="Maroon" title="c10">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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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="66_2_1_2_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c16">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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font></span>)</span> implies <font color="Maroon" title="c10">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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="c16">a</font> )</span><br/>

<b>assume </b><a NAME="E1:66_2_1_2_7"/>
<font color="Maroon" title="c16">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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="c4">B</font></span>)</span>,<font color="Maroon" title="c14">l</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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="c16">a</font></span><br/>

<a NAME="E2:66_2_1_2_7"/><b>then </b><i><font color="Green" title="E61">A125</font></i>: 
<font color="Maroon" title="c16">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_2_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_2_1_2_7_1"/>
( <font color="Maroon" title="c16">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="c16">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="66_2_1_2_7_1_1"><b>suppose </b></a><a NAME="E1:66_2_1_2_7_1_1"/><i><font color="Green" title="E62">A126</font></i>: 
<font color="Maroon" title="c16">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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="c16">a</font></span><br/><div class="add"><a NAME="E2:66_2_1_2_7_1_1"/><b>then </b>
<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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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="c16">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">l</font>
 <b>by </b><i><a class="txt" href="#E22:66_2_1_2"><i><font color="Green" title="E55">A113</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><b>hence </b><a NAME="E3:66_2_1_2_7_1_1"/>
<font color="Maroon" title="c10">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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="c16">a</font>
 <b>by </b><i><a class="txt" href="#E21:66_2_1_2"><i><font color="Green" title="E54">A112</font></i></a>, <a class="txt" href="#E1:66_2_1_2_7_1_1"><i><font color="Green" title="E62">A126</font></i></a>, <a class="ref" href="zfmisc_1.html#T38" title="ZFMISC_1:th.38">ZFMISC_1:38</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_2_7_1_2"><b>suppose </b></a><a NAME="E1:66_2_1_2_7_1_2"/><i><font color="Green" title="E62">A127</font></i>: 
<font color="Maroon" title="c16">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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="c16">a</font></span><br/><div class="add"><a NAME="E2:66_2_1_2_7_1_2"/>
<font color="Maroon" title="c8">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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="c8">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">a</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:66_2_1_2_7_1_2"/><b>then </b>
<font color="Maroon" title="c8">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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="c8">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="c16">a</font>
 <b>by </b><i><a class="txt" href="#E2:66_2_1_2_7"><i><font color="Green" title="E61">A125</font></i></a>, <a class="ref" href="pzfmisc1.html#D1" title="PZFMISC1:def.1">PZFMISC1:def 1</a></i>;<br/><a NAME="E4:66_2_1_2_7_1_2"/><b>then </b>
<font color="Maroon" title="c10">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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="c8">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="c16">a</font>
 <b>by </b><i><a class="txt" href="#E4:66"><i><font color="Green" title="E45">A102</font></i></a>, <a class="txt" href="#E1:66_2"><i><font color="Green" title="E47">A104</font></i></a>, <a class="txt" href="#E3:66_2_1_2"><i><font color="Green" title="E49">A106</font></i></a>, <a class="txt" href="#E6:66_2_1_2"><i><font color="Green" title="E50">A107</font></i></a>, <a class="txt" href="#E1:66_2_1_2_7_1_2"><i><font color="Green" title="E62">A127</font></i></a>, <a class="ref" href="pencil_1.html#T23" target="_self" title="PENCIL_1:th.23">Th23</a></i>;<br/><b>hence </b><a NAME="E5:66_2_1_2_7_1_2"/>
<font color="Maroon" title="c10">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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="c8">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K1" title="POLYNOM1: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="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="c16">a</font>
 <b>by </b><i><a class="txt" href="#E1:66_2_1_2_7_1_2"><i><font color="Green" title="E62">A127</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E34:66_2_1_2"/><b>then </b>
<font color="Maroon" title="c10">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">L</font>
 <b>by </b><i><a class="txt" href="#E32:66_2_1_2"><i><font color="Green" title="E60">A124</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><a NAME="E35:66_2_1_2"/><b>then </b>
<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>
 <b>by </b><i><a class="txt" href="#E3:66_2_1_2"><i><font color="Green" title="E49">A106</font></i></a>, <a class="txt" href="#E6:66_2_1_2"><i><font color="Green" title="E50">A107</font></i></a>, <a class="txt" href="#E31:66_2_1_2"><i><font color="Green" title="E59">A123</font></i></a>, <a class="ref" href="zfmisc_1.html#T38" title="ZFMISC_1:th.38">ZFMISC_1:38</a></i>;<br/><b>hence </b><a NAME="E36:66_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> 
 <b>by </b><i><a class="ref" href="pencil_1.html#D1" target="_self" title="PENCIL_1:def.1">Def1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E8:66"/>
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_3"><b>proof </b></a><div class="add">

<b>let </b><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#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>assume </b><a NAME="E1:66_3"/><i><font color="Green" title="E47">A128</font></i>: 
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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>then consider </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:66_3"/><i><font color="Green" title="E48">A129</font></i>: 
( <font color="Maroon" title="c6">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> &amp; <font color="Maroon" title="c7">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> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b</font> )
 <b>by </b><i><a class="ref" href="pencil_1.html#T2" target="_self" title="PENCIL_1:th.2">Th2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">a1</font> = <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c9">b1</font> = <font color="Maroon" title="c7">b</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="txt" href="#E3:66_3"><i><font color="Green" title="E48">A129</font></i></a>, <a class="ref" href="pencil_1.html#T14" target="_self" title="PENCIL_1:th.14">Th14</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">L</font> being  <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>  <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E6:66_3"/><i><font color="Green" title="E49">A130</font></i>: 
( <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="c10">L</font> &amp;  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="c10">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> )
 <b>by </b><i><a class="ref" href="pencil_1.html#D22" target="_self" title="PENCIL_1:def.22">Def22</a></i>;<br/>
<a NAME="E7:66_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>
 
<b>by </b><i><a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E8:66_3"/><b>then </b>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c5">l</font>
 
<b>by </b><i><a class="txt" href="#E1:66_3"><i><font color="Green" title="E47">A128</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">L</font> = <font color="Maroon" title="c10">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#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="txt" href="#E6:66_3"><i><font color="Green" title="E49">A130</font></i></a>, <a class="ref" href="pencil_1.html#T13" target="_self" title="PENCIL_1:th.13">Th13</a></i>;<br/>
<a NAME="E10:66_3"/><i><font color="Green" title="E50">A131</font></i>: 
(  <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="c11">L</font> &amp; ( 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#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/><font color="Maroon" title="c4">B</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="c11">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) )
 
<b>by </b><i><a class="txt" href="#E4:66"><i><font color="Green" title="E45">A102</font></i></a>, <a class="txt" href="#E1:66_3"><i><font color="Green" title="E47">A128</font></i></a>, <a class="txt" href="#E6:66_3"><i><font color="Green" title="E49">A130</font></i></a>, <a class="ref" href="pencil_1.html#T26" target="_self" title="PENCIL_1:th.26">Th26</a></i>;<br/>
<i><font color="Green" title="E51">A132</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">L</font> = 
<font color="Maroon" title="c1">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">B</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/>
<a NAME="E12:66_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="c11">L</font> holds <br/><font color="Maroon" title="c11">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="66_3_2"><b>proof </b></a><div class="add">

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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="c11">L</font> implies <font color="Maroon" title="c11">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/>

<b>assume </b><a NAME="E1:66_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="c11">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">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="66_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_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="66_3_2_1_1"><b>suppose </b></a><a NAME="E1:66_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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">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"><b>hence </b><a NAME="E2:66_3_2_1_1"/>
<font color="Maroon" title="c11">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>
 <b>by </b><i><a class="txt" href="#E4:66"><i><font color="Green" title="E45">A102</font></i></a>, <a class="txt" href="#E1:66_3"><i><font color="Green" title="E47">A128</font></i></a>, <a class="txt" href="#E6:66_3"><i><font color="Green" title="E49">A130</font></i></a>, <a class="ref" href="pencil_1.html#T26" target="_self" title="PENCIL_1:th.26">Th26</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_3_2_1_2"><b>suppose </b></a><a NAME="E1:66_3_2_1_2"/><i><font color="Green" title="E52">A133</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">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"><b>consider </b><font color="Maroon" title="c13">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font><b> such that </b><br/><a NAME="E3:66_3_2_1_2"/><i><font color="Green" title="E53">A134</font></i>: 
<font color="Maroon" title="c11">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">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="c13">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E6:66_3"><i><font color="Green" title="E49">A130</font></i></a></i>;<br/><a NAME="E4:66_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="c11">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:66_3_2_1_2"><i><font color="Green" title="E53">A134</font></i></a>, <a class="ref" href="pencil_1.html#D6" target="_self" title="PENCIL_1:def.6">Def6</a></i>;<br/><a NAME="E5:66_3_2_1_2"/><b>then </b>
not <font color="Maroon" title="c11">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>
 <b>by </b><i><a class="ref" href="pencil_1.html#T4" target="_self" title="PENCIL_1:th.4">Th4</a></i>;<br/><a NAME="E6:66_3_2_1_2"/><b>then </b>
<font color="Maroon" title="c13">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="c11">L</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D21" target="_self" title="PENCIL_1:def.21">Def21</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c14">Li</font> = <font color="Maroon" title="c11">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> <b>by </b><i><a class="txt" href="#E10:66_3"><i><font color="Green" title="E50">A131</font></i></a>, <a class="txt" href="#E3:66_3_2_1_2"><i><font color="Green" title="E53">A134</font></i></a></i>;<br/><a NAME="E8:66_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>
 <b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/><a NAME="E9:66_3_2_1_2"/><b>then </b>
<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>
 <b>by </b><i><a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c15">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> <b>by </b><i><a class="ref" href="yellow_6.html#T9" title="YELLOW_6:th.9">YELLOW_6:9</a></i>;<br/><a NAME="E11:66_3_2_1_2"/><i><font color="Green" title="E54">A135</font></i>: 
( <font color="Maroon" title="c8">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> &amp; <font color="Maroon" title="c9">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> )
 <b>by </b><i><a class="txt" href="#E4:66"><i><font color="Green" title="E45">A102</font></i></a>, <a class="txt" href="#E3:66_3"><i><font color="Green" title="E48">A129</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><div><i><font color="Green" title="E55">A136</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_3_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_3_2_1_2_1"/><i><font color="Green" title="E56">A137</font></i>: 
<font color="Maroon" title="c8">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="c9">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:66_3_2_1_2_1"/><i><font color="Green" title="E57">A138</font></i>: 
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="c8">a1</font> holds <br/><font color="Maroon" title="c8">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="c9">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="66_3_2_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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="c8">a1</font> implies <font color="Maroon" title="c8">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="c9">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">o</font> )</span><br/>

<b>assume </b><a NAME="E1:66_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="c8">a1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">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="c9">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="66_3_2_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_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="66_3_2_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:66_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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">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="c9">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"><b>hence </b><a NAME="E2:66_3_2_1_2_1_1_1_1"/>
<font color="Maroon" title="c8">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="c9">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">o</font>
 <b>by </b><i><a class="txt" href="#E11:66_3_2_1_2"><i><font color="Green" title="E54">A135</font></i></a>, <a class="ref" href="pencil_1.html#T23" target="_self" title="PENCIL_1:th.23">Th23</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_3_2_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:66_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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">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="c9">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"><b>hence </b><a NAME="E2:66_3_2_1_2_1_1_1_2"/>
<font color="Maroon" title="c8">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="c9">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">o</font>
 <b>by </b><i><a class="txt" href="#E1:66_3_2_1_2_1"><i><font color="Green" title="E56">A137</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div> <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">a1</font> = 
<font color="Maroon" title="c1">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">b1</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/><b>hence </b><a NAME="E4:66_3_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:66_3"><i><font color="Green" title="E48">A129</font></i></a>, <a class="txt" href="#E2:66_3_2_1_2_1"><i><font color="Green" title="E57">A138</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E13:66_3_2_1_2"/>
( <font color="Maroon" title="c8">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="c11">L</font> &amp; <font color="Maroon" title="c9">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="c11">L</font> )
 <b>by </b><i><a class="txt" href="#E3:66_3"><i><font color="Green" title="E48">A129</font></i></a>, <a class="txt" href="#E6:66_3"><i><font color="Green" title="E49">A130</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E14:66_3_2_1_2"/><b>then </b><i><font color="Green" title="E56">A139</font></i>: 
( <font color="Maroon" title="c8">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="c11">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> &amp; <font color="Maroon" title="c9">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="c11">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> )
 <b>by </b><i><a class="txt" href="#E6:66"><i><font color="Green" title="E46">A103</font></i></a>, <a class="txt" href="#E11:66_3"><i><font color="Green" title="E51">A132</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><a NAME="E15:66_3_2_1_2"/>
( <font color="Maroon" title="c8">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> &amp; <font color="Maroon" title="c9">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> )
 <b>by </b><i><a class="txt" href="#E6:66"><i><font color="Green" title="E46">A103</font></i></a>, <a class="txt" href="#E11:66_3_2_1_2"><i><font color="Green" title="E54">A135</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><a NAME="E16:66_3_2_1_2"/><b>then </b>
( <font color="Maroon" title="c8">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="c14">Li</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c15">C</font> &amp; <font color="Maroon" title="c9">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="c14">Li</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c15">C</font> )
 <b>by </b><i><a class="txt" href="#E14:66_3_2_1_2"><i><font color="Green" title="E56">A139</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E17:66_3_2_1_2"/><b>then </b><i><font color="Green" title="E57">A140</font></i>: 
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="c14">Li</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c15">C</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E12:66_3_2_1_2"><i><font color="Green" title="E55">A136</font></i></a>, <a class="ref" href="pencil_1.html#T2" target="_self" title="PENCIL_1:th.2">Th2</a></i>;<br/><a NAME="E18:66_3_2_1_2"/>
<font color="Maroon" title="c15">C</font> is <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a>
 <b>by </b><i><a class="txt" href="#E4:66"><i><font color="Green" title="E45">A102</font></i></a></i>;<br/><b>hence </b><a NAME="E19:66_3_2_1_2"/>
<font color="Maroon" title="c11">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>
 <b>by </b><i><a class="txt" href="#E1:66_3_2_1_2"><i><font color="Green" title="E52">A133</font></i></a>, <a class="txt" href="#E17:66_3_2_1_2"><i><font color="Green" title="E57">A140</font></i></a>, <a class="ref" href="pencil_1.html#D2" target="_self" title="PENCIL_1:def.2">Def2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E13:66_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>
 <b>by </b><i><a class="txt" href="#E4:66"><i><font color="Green" title="E45">A102</font></i></a>, <a class="txt" href="#E6:66_3"><i><font color="Green" title="E49">A130</font></i></a>, <a class="txt" href="#E11:66_3"><i><font color="Green" title="E51">A132</font></i></a>, <a class="ref" href="card_3.html#T38" title="CARD_3:th.38">CARD_3:38</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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


</div>
