<?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> ;<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>;<br/><b>let </b><font color="Maroon" title="c3">B1</font>, <font color="Maroon" title="c4">B2</font> be    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font>;<br/>





<b>assume </b><a NAME="E1:23"/><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon" title="c3">B1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">B2</font>
 ;<br/>

<b>thus </b><a NAME="E2:23"/>
( <font color="Maroon" title="c3">B1</font> <a href="pencil_3.html#R1" title="PENCIL_3:pred.1">'||'</a> <font color="Maroon" title="c4">B2</font> implies for <font color="Olive" title="b1">b1</font>, <font color="Olive" title="b2">b2</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 <font color="Maroon" title="c3">B1</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">b1</font> &amp; <font color="Maroon" title="c4">B2</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">b2</font> holds <br/>(  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Olive" title="b1">b1</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="Olive" title="b2">b2</font> &amp;  ex <font color="Olive" title="b3">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> st <br/>( <font color="Olive" title="b3">r</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="Olive" title="b1">b1</font> &amp; ( for <font color="Olive" title="b4">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="b4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">r</font> holds <br/><font color="Olive" title="b1">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> ) &amp; ( for <font color="Olive" title="b4">c1</font>, <font color="Olive" title="b5">c2</font> being  <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> <font color="Olive" title="b3">r</font></span>)</span>  st <font color="Olive" title="b1">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b4">c1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Olive" title="b2">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b5">c2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds <br/><font color="Olive" title="b4">c1</font>,<font color="Olive" title="b5">c2</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a>  ) ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1"/><i><font color="Green" title="E23">A2</font></i>: 
<font color="Maroon" title="c3">B1</font> <a href="pencil_3.html#R1" title="PENCIL_3:pred.1">'||'</a> <font color="Maroon" title="c4">B2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">b1</font>, <font color="Maroon" title="c6">b2</font> be  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>;<br/>

<b>assume </b><a NAME="E2:23_1"/><i><font color="Green" title="E24">A3</font></i>: 
( <font color="Maroon" title="c3">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c5">b1</font> &amp; <font color="Maroon" title="c4">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c6">b2</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c7">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="E4:23_1"/><i><font color="Green" title="E25">A4</font></i>: 
( <font color="Maroon" title="c3">B1</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="c7">L1</font> &amp; <font color="Maroon" title="c7">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="c7">L1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c7">L1</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_2.html#D2" title="PENCIL_2:def.2">PENCIL_2:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">L2</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="E6:23_1"/><i><font color="Green" title="E26">A5</font></i>: 
( <font color="Maroon" title="c4">B2</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="c8">L2</font> &amp; <font color="Maroon" title="c8">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="c8">L2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c8">L2</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_2.html#D2" title="PENCIL_2:def.2">PENCIL_2:def 2</a></i>;<br/>
<a NAME="E7:23_1"/><i><font color="Green" title="E27">A6</font></i>: 
<font color="Maroon" title="c5">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c7">L1</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E2:23_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="pencil_3.html#E4:23_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="pua2mss1.html#T2" title="PUA2MSS1:th.2">PUA2MSS1:2</a></i>;<br/>
<a NAME="E8:23_1"/><i><font color="Green" title="E28">A7</font></i>: 
<font color="Maroon" title="c6">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c8">L2</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E2:23_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="pencil_3.html#E6:23_1"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="pua2mss1.html#T2" title="PUA2MSS1:th.2">PUA2MSS1:2</a></i>;<br/>
<b>thus </b><a NAME="E9:23_1"/><i><font color="Green" title="E29">A8</font></i>: 
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">b1</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="c6">b2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1_1"/>
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">b1</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="c6">b2</font>
 ;<br/>

<a NAME="E2:23_1_1"/><b>then </b>
( not <font color="Maroon" title="c6">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">b1</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c6">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">b1</font></span>)</span> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> )
 
<b>by </b><i><a class="ref" href="pencil_1.html#T12" title="PENCIL_1:th.12">PENCIL_1:12</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">c2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:23_1_1"/><i><font color="Green" title="E30">A9</font></i>: 
<font color="Maroon" title="c6">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">b1</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="c9">c2</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/>
<b>consider </b><font color="Maroon" title="c10">p0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:23_1_1"/><i><font color="Green" title="E31">A10</font></i>: 
<font color="Maroon" title="c10">p0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:23_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">p0</font> = <font color="Maroon" title="c10">p0</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="pencil_3.html#E6:23_1_1"><i><font color="Green" title="E31">A10</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">p</font> = <font color="Maroon" title="c11">p0</font> as    <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</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_3.html#T6" target="_self" title="PENCIL_3:th.6">Th6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">bl</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="c5">b1</font></span>)</span></span>)</span>;<br/>
<a NAME="E10:23_1_1"/><i><font color="Green" title="E32">A11</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> <font color="Maroon" title="c13">bl</font>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D6" title="PENCIL_1:def.6">PENCIL_1:def 6</a></i>;<br/>
<a NAME="E11:23_1_1"/>
<font color="Maroon" title="c13">bl</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="c5">b1</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E12:23_1_1"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c13">bl</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> 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="c5">b1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></i>;<br/>
<a NAME="E13:23_1_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> 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="c5">b1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E10:23_1_1"><i><font color="Green" title="E32">A11</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E15:23_1_1"/><i><font color="Green" title="E33">A12</font></i>: 
( <font color="Maroon" title="c14">a</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> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">b1</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c14">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c2</font> )
 <b>by </b><i><a class="ref" href="pencil_1.html#T3" title="PENCIL_1:th.3">PENCIL_1:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">a</font> = <font color="Maroon" title="c14">a</font> 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="c5">b1</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="pencil_3.html#E15:23_1_1"><i><font color="Green" title="E33">A12</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">x</font> = <font color="Maroon" title="c12">p</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">b1</font></span>)</span>,<font color="Maroon" title="c15">a</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" title="PENCIL_1:th.25">PENCIL_1:25</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">x1</font> = <font color="Maroon" title="c16">x</font> as    <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</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_3.html#T6" target="_self" title="PENCIL_3:th.6">Th6</a></i>;<br/>
<i><font color="Green" title="E34">A13</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c17">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#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">b1</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="23_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:23_1_1_2"/><i><font color="Green" title="E35">A14</font></i>: 
<font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c17">x1</font>
 ;<br/><a NAME="E2:23_1_1_2"/><b>then </b>
<font color="Maroon" title="c18">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="E3:23_1_1_2"/><b>then </b><i><font color="Green" title="E36">A15</font></i>: 
<font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c12">p</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="23_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1_2_1"/>
( <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">b1</font> or <font color="Maroon" title="c18">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">b1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:23_1_1_2_1_1"/><i><font color="Green" title="E37">A16</font></i>: 
<font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">b1</font>
 ;<br/><div class="add"><a NAME="E2:23_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c17">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">a</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:23_1_1_2"><i><font color="Green" title="E36">A15</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="E3:23_1_1_2_1_1"/>
<font color="Maroon" title="c17">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E4:23_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="pencil_3.html#E7:23_1"><i><font color="Green" title="E27">A6</font></i></a>, <a class="txt" href="pencil_3.html#E1:23_1_1_2_1_1"><i><font color="Green" title="E37">A16</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:23_1_1_2_1_2"/><i><font color="Green" title="E37">A17</font></i>: 
<font color="Maroon" title="c18">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">b1</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c19">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:23_1_1_2_1_2"/><i><font color="Green" title="E38">A18</font></i>: 
( <font color="Maroon" title="c19">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c19">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">b1</font> &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#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">b1</font> holds <br/><font color="Maroon" title="c19">f</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> <font color="Maroon" title="c5">b1</font> <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="txt" href="pencil_3.html#E2:23_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="pencil_3.html#E6:23_1_1"><i><font color="Green" title="E31">A10</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="E4:23_1_1_2_1_2"/>
<font color="Maroon" title="c17">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:23_1_1_2_1_2"><i><font color="Green" title="E37">A17</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><a NAME="E5:23_1_1_2_1_2"/>
<font color="Maroon" title="c17">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E19:23_1_1"><i><font color="Green" title="E34">A13</font></i></a>, <a class="txt" href="pencil_3.html#E1:23_1_1_2"><i><font color="Green" title="E35">A14</font></i></a>, <a class="txt" href="pencil_3.html#E3:23_1_1_2_1_2"><i><font color="Green" title="E38">A18</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E21:23_1_1"/><b>then </b><i><font color="Green" title="E35">A19</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E2:23_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="pencil_3.html#E19:23_1_1"><i><font color="Green" title="E34">A13</font></i></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c18">y</font> being   <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> such that </b><br/><a NAME="E23:23_1_1"/><i><font color="Green" title="E36">A20</font></i>: 
( <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B2</font> &amp; <font color="Maroon" title="c16">x</font>,<font color="Maroon" title="c18">y</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a>  )
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:23_1"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="pencil_3.html#D2" target="_self" title="PENCIL_3:def.2">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">y1</font> = <font color="Maroon" title="c18">y</font> as    <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</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_3.html#T6" target="_self" title="PENCIL_3:th.6">Th6</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1_3"/>
( <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font> or <font color="Maroon" title="c18">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3_1"><b>suppose </b></a><a NAME="E1:23_1_1_3_1"/>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font>
 ;<br/><div class="add"><a NAME="E2:23_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B2</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E21:23_1_1"><i><font color="Green" title="E35">A19</font></i></a>, <a class="txt" href="pencil_3.html#E23:23_1_1"><i><font color="Green" title="E36">A20</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:23_1_1_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:23"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3_2"><b>suppose </b></a><a NAME="E1:23_1_1_3_2"/>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">x</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c20">i0</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="E3:23_1_1_3_2"/><i><font color="Green" title="E37">A21</font></i>: 
( ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <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> <font color="Maroon" title="c20">i0</font></span>)</span>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i0</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i0</font> holds <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a>  ) ) &amp; ( for <font color="Olive" title="b1">j</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">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c20">i0</font> holds <br/><font color="Maroon" title="c17">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> ) )
 <b>by </b><i><a class="txt" href="pencil_3.html#E23:23_1_1"><i><font color="Green" title="E36">A20</font></i></a>, <a class="ref" href="pencil_3.html#T17" target="_self" title="PENCIL_3:th.17">Th17</a></i>;<br/><i><font color="Green" title="E38">A22</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c19">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#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">b1</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="23_1_1_3_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c21">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:23_1_1_3_2_2"/><i><font color="Green" title="E39">A23</font></i>: 
<font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c19">y1</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c22">i5</font> = <font color="Maroon" title="c21">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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="23_1_1_3_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1_3_2_2_1"/>
( <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">b1</font> or <font color="Maroon" title="c21">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">b1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3_2_2_1_1"><b>suppose </b></a><a NAME="E1:23_1_1_3_2_2_1_1"/><i><font color="Green" title="E40">A24</font></i>: 
<font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">b1</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c23">i1</font> = <font color="Maroon" title="c21">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="txt" href="pencil_3.html#E1:23_1_1_3_2_2"><i><font color="Green" title="E39">A23</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:23_1_1_3_2_2_1_1"/>
<font color="Maroon" title="c19">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i1</font>
 <b>by </b><i><a class="ref" href="pboole.html#D17" title="PBOOLE:def.17">PBOOLE:def 17</a></i>;<br/><a NAME="E4:23_1_1_3_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c19">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i1</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <font color="Maroon" title="c23">i1</font></span>)</span>
 <b>by </b><i><a class="ref" href="pencil_3.html#T7" target="_self" title="PENCIL_3:th.7">Th7</a></i>;<br/><b>hence </b><a NAME="E5:23_1_1_3_2_2_1_1"/>
<font color="Maroon" title="c19">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E4:23_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="pencil_3.html#E7:23_1"><i><font color="Green" title="E27">A6</font></i></a>, <a class="txt" href="pencil_3.html#E1:23_1_1_3_2_2_1_1"><i><font color="Green" title="E40">A24</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3_2_2_1_2"><b>suppose </b></a><a NAME="E1:23_1_1_3_2_2_1_2"/><i><font color="Green" title="E40">A25</font></i>: 
<font color="Maroon" title="c21">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">b1</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c23">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:23_1_1_3_2_2_1_2"/><i><font color="Green" title="E41">A26</font></i>: 
( <font color="Maroon" title="c23">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">b1</font> &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#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">b1</font> holds <br/><font color="Maroon" title="c23">f</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> <font color="Maroon" title="c5">b1</font> <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="txt" href="pencil_3.html#E2:23_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="pencil_3.html#E6:23_1_1"><i><font color="Green" title="E31">A10</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="E4:23_1_1_3_2_2_1_2"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c12">p</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="E5:23_1_1_3_2_2_1_2"/><b>then </b><i><font color="Green" title="E42">A27</font></i>: 
<font color="Maroon" title="c17">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="c5">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">a</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><b>consider </b><font color="Maroon" title="c24">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E7:23_1_1_3_2_2_1_2"/><i><font color="Green" title="E43">A28</font></i>: 
( <font color="Maroon" title="c24">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y1</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c24">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">b2</font> &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#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">b2</font> holds <br/><font color="Maroon" title="c24">g</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> <font color="Maroon" title="c6">b2</font> <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="txt" href="pencil_3.html#E2:23_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="pencil_3.html#E23:23_1_1"><i><font color="Green" title="E36">A20</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="E8:23_1_1_3_2_2_1_2"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">b2</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="E9:23_1_1_3_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c19">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="c5">b1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">b1</font></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E7:23_1_1_3_2_2_1_2"><i><font color="Green" title="E43">A28</font></i></a></i>;<br/><a NAME="E10:23_1_1_3_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c19">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="c5">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">c2</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E4:23_1_1"><i><font color="Green" title="E30">A9</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E11:23_1_1_3_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c20">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">b1</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E15:23_1_1"><i><font color="Green" title="E33">A12</font></i></a>, <a class="txt" href="pencil_3.html#E3:23_1_1_3_2"><i><font color="Green" title="E37">A21</font></i></a>, <a class="txt" href="pencil_3.html#E5:23_1_1_3_2_2_1_2"><i><font color="Green" title="E42">A27</font></i></a></i>;<br/><a NAME="E12:23_1_1_3_2_2_1_2"/><b>then </b><i><font color="Green" title="E44">A29</font></i>: 
<font color="Maroon" title="c19">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i5</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:23_1_1_3_2"><i><font color="Green" title="E37">A21</font></i></a>, <a class="txt" href="pencil_3.html#E1:23_1_1_3_2_2_1_2"><i><font color="Green" title="E40">A25</font></i></a></i>;<br/><a NAME="E13:23_1_1_3_2_2_1_2"/>
<font color="Maroon" title="c17">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:23_1_1_3_2_2_1_2"><i><font color="Green" title="E40">A25</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><a NAME="E14:23_1_1_3_2_2_1_2"/>
<font color="Maroon" title="c19">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E4:23_1_1_3_2"><i><font color="Green" title="E38">A22</font></i></a>, <a class="txt" href="pencil_3.html#E1:23_1_1_3_2_2"><i><font color="Green" title="E39">A23</font></i></a>, <a class="txt" href="pencil_3.html#E3:23_1_1_3_2_2_1_2"><i><font color="Green" title="E41">A26</font></i></a>, <a class="txt" href="pencil_3.html#E12:23_1_1_3_2_2_1_2"><i><font color="Green" title="E44">A29</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E6:23_1_1_3_2"/><b>then </b>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:23_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="pencil_3.html#E4:23_1_1_3_2"><i><font color="Green" title="E38">A22</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="E7:23_1_1_3_2"/><b>then </b>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B2</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E23:23_1_1"><i><font color="Green" title="E36">A20</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E8:23_1_1_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:23"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E10:23_1"/>
 ex <font color="Olive" title="b1">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> st <br/>( <font color="Olive" title="b1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">b1</font> &amp; ( 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> <font color="Olive" title="b1">r</font> holds <br/><font color="Maroon" title="c5">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> ) &amp; ( for <font color="Olive" title="b2">c1</font>, <font color="Olive" title="b3">c2</font> being  <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> <font color="Olive" title="b1">r</font></span>)</span>  st <font color="Maroon" title="c5">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b2">c1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c6">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b3">c2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds <br/><font color="Olive" title="b2">c1</font>,<font color="Olive" title="b3">c2</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a>  ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c9">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:23_1_2"/><i><font color="Green" title="E30">A30</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:23_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">x</font> = <font color="Maroon" title="c9">x</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="pencil_3.html#E2:23_1_2"><i><font color="Green" title="E30">A30</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">x1</font> = <font color="Maroon" title="c10">x</font> as    <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</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_3.html#T6" target="_self" title="PENCIL_3:th.6">Th6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">y</font> being   <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> such that </b><br/><a NAME="E6:23_1_2"/><i><font color="Green" title="E31">A31</font></i>: 
( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B2</font> &amp; <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c12">y</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a>  )
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:23_1"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="pencil_3.html#E2:23_1_2"><i><font color="Green" title="E30">A30</font></i></a>, <a class="ref" href="pencil_3.html#D2" target="_self" title="PENCIL_3:def.2">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">y1</font> = <font color="Maroon" title="c12">y</font> as    <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</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_3.html#T6" target="_self" title="PENCIL_3:th.6">Th6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_2_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y</font>
 ;<br/><a NAME="E2:23_1_2_1"/><b>then </b>
<font color="Maroon" title="c3">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:23_1_2"><i><font color="Green" title="E30">A30</font></i></a>, <a class="txt" href="pencil_3.html#E6:23_1_2"><i><font color="Green" title="E31">A31</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:23_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:23"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c14">r</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="E10:23_1_2"/><i><font color="Green" title="E32">A32</font></i>: 
( ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <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> <font color="Maroon" title="c14">r</font></span>)</span>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">r</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">r</font> holds <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a>  ) ) &amp; ( for <font color="Olive" title="b1">j</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">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">r</font> holds <br/><font color="Maroon" title="c11">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> ) )
 <b>by </b><i><a class="txt" href="pencil_3.html#E6:23_1_2"><i><font color="Green" title="E31">A31</font></i></a>, <a class="ref" href="pencil_3.html#T17" target="_self" title="PENCIL_3:th.17">Th17</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c14">r</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_2_2"/><i><font color="Green" title="E33">A33</font></i>: 
<font color="Maroon" title="c14">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">b1</font>
 ;<br/><i><font color="Green" title="E34">A34</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c13">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#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">b1</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="23_1_2_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:23_1_2_2_2"/><i><font color="Green" title="E35">A35</font></i>: 
<font color="Maroon" title="c15">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">b1</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c16">o1</font> = <font color="Maroon" title="c15">o</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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="23_1_2_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_2_2_2_1"/>
( <font color="Maroon" title="c16">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">r</font> or <font color="Maroon" title="c16">o1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">r</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_2_2_1_1"><b>suppose </b></a><a NAME="E1:23_1_2_2_2_1_1"/><i><font color="Green" title="E36">A36</font></i>: 
<font color="Maroon" title="c16">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">r</font>
 ;<br/><div class="add"><a NAME="E2:23_1_2_2_2_1_1"/>
<font color="Maroon" title="c13">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">o1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">o1</font>
 <b>by </b><i><a class="ref" href="pboole.html#D17" title="PBOOLE:def.17">PBOOLE:def 17</a></i>;<br/><a NAME="E3:23_1_2_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c13">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">o1</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <font color="Maroon" title="c16">o1</font></span>)</span>
 <b>by </b><i><a class="ref" href="pencil_3.html#T7" target="_self" title="PENCIL_3:th.7">Th7</a></i>;<br/><b>hence </b><a NAME="E4:23_1_2_2_2_1_1"/>
<font color="Maroon" title="c13">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E4:23_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="pencil_3.html#E7:23_1"><i><font color="Green" title="E27">A6</font></i></a>, <a class="txt" href="pencil_3.html#E1:23_1_2_2"><i><font color="Green" title="E33">A33</font></i></a>, <a class="txt" href="pencil_3.html#E1:23_1_2_2_2_1_1"><i><font color="Green" title="E36">A36</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_2_2_1_2"><b>suppose </b></a><a NAME="E1:23_1_2_2_2_1_2"/><i><font color="Green" title="E36">A37</font></i>: 
<font color="Maroon" title="c16">o1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">r</font>
 ;<br/><div class="add"><a NAME="E2:23_1_2_2_2_1_2"/><b>then </b>
( <font color="Maroon" title="c5">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">o1</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> &amp; not <font color="Maroon" title="c5">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">o1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:23_1_2_2"><i><font color="Green" title="E33">A33</font></i></a>, <a class="ref" href="pencil_1.html#T12" title="PENCIL_1:th.12">PENCIL_1:12</a></i>;<br/><b>then consider </b><font color="Maroon" title="c17">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:23_1_2_2_2_1_2"/><i><font color="Green" title="E37">A38</font></i>: 
<font color="Maroon" title="c5">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">o1</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="c17">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="E5:23_1_2_2_2_1_2"/>
<font color="Maroon" title="c11">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">o1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">o1</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:23_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="pencil_3.html#E2:23_1_2"><i><font color="Green" title="E30">A30</font></i></a>, <a class="txt" href="pencil_3.html#E1:23_1_2_2_2"><i><font color="Green" title="E35">A35</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><b>then </b><font color="Maroon" title="c17">c</font> = 
<font color="Maroon" title="c11">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">o1</font>
<b>by </b><i><a class="txt" href="pencil_3.html#E4:23_1_2_2_2_1_2"><i><font color="Green" title="E37">A38</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c13">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">o1</font>
<b>by </b><i><a class="txt" href="pencil_3.html#E10:23_1_2"><i><font color="Green" title="E32">A32</font></i></a>, <a class="txt" href="pencil_3.html#E1:23_1_2_2_2_1_2"><i><font color="Green" title="E36">A37</font></i></a></i>
;<br/><b>hence </b><a NAME="E7:23_1_2_2_2_1_2"/>
<font color="Maroon" title="c13">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E4:23_1_2_2_2_1_2"><i><font color="Green" title="E37">A38</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E4:23_1_2_2"/><b>then </b>
<font color="Maroon" title="c13">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:23_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="pencil_3.html#E2:23_1_2_2"><i><font color="Green" title="E34">A34</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><a NAME="E5:23_1_2_2"/><b>then </b>
<font color="Maroon" title="c3">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="pencil_3.html#E6:23_1_2"><i><font color="Green" title="E31">A31</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E6:23_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:23"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:23_1_2"/>
<font color="Maroon" title="c14">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">b1</font>
 ;<br/>

<b>thus </b><a NAME="E13:23_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> <font color="Maroon" title="c14">r</font> holds <br/><font color="Maroon" title="c5">b1</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="c6">b2</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="23_1_2_3"><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>;<br/>

<b>assume </b><a NAME="E1:23_1_2_3"/><i><font color="Green" title="E33">A39</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">r</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_2_3_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="c5">b1</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="c5">b1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:23_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="c5">b1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:23_1_2_3_1_1"/>
<font color="Maroon" title="c5">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> <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="pencil_3.html#E4:23_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="pencil_3.html#E6:23_1"><i><font color="Green" title="E26">A5</font></i></a>, <a class="txt" href="pencil_3.html#E7:23_1"><i><font color="Green" title="E27">A6</font></i></a>, <a class="txt" href="pencil_3.html#E8:23_1"><i><font color="Green" title="E28">A7</font></i></a>, <a class="txt" href="pencil_3.html#E9:23_1"><i><font color="Green" title="E29">A8</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:23_1_2_3_1_2"/><i><font color="Green" title="E34">A40</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">b1</font>
 ;<br/><div class="add"><a NAME="E2:23_1_2_3_1_2"/><b>then </b>
( <font color="Maroon" title="c5">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> &amp; not <font color="Maroon" title="c5">b1</font> <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> )
 <b>by </b><i><a class="ref" href="pencil_1.html#T12" title="PENCIL_1:th.12">PENCIL_1:12</a></i>;<br/><b>then consider </b><font color="Maroon" title="c16">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:23_1_2_3_1_2"/><i><font color="Green" title="E35">A41</font></i>: 
<font color="Maroon" title="c5">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#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="c16">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="E5:23_1_2_3_1_2"/>
( <font color="Maroon" title="c6">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> &amp; not <font color="Maroon" title="c6">b2</font> <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> )
 <b>by </b><i><a class="txt" href="pencil_3.html#E9:23_1"><i><font color="Green" title="E29">A8</font></i></a>, <a class="txt" href="pencil_3.html#E1:23_1_2_3_1_2"><i><font color="Green" title="E34">A40</font></i></a>, <a class="ref" href="pencil_1.html#T12" title="PENCIL_1:th.12">PENCIL_1:12</a></i>;<br/><b>then consider </b><font color="Maroon" title="c17">d</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:23_1_2_3_1_2"/><i><font color="Green" title="E36">A42</font></i>: 
<font color="Maroon" title="c6">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c17">d</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="E8:23_1_2_3_1_2"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">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="E9:23_1_2_3_1_2"/><b>then </b>
<font color="Maroon" title="c11">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> <font color="Maroon" title="c5">b1</font> <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="pencil_3.html#E2:23_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="pencil_3.html#E2:23_1_2"><i><font color="Green" title="E30">A30</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><b>then </b><i><font color="Green" title="E37">A43</font></i>: <font color="Maroon" title="c16">c</font> = 
<font color="Maroon" title="c11">x1</font> <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="pencil_3.html#E4:23_1_2_3_1_2"><i><font color="Green" title="E35">A41</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c13">y1</font> <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="pencil_3.html#E10:23_1_2"><i><font color="Green" title="E32">A32</font></i></a>, <a class="txt" href="pencil_3.html#E1:23_1_2_3"><i><font color="Green" title="E33">A39</font></i></a></i>
;<br/><a NAME="E11:23_1_2_3_1_2"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">b2</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="E12:23_1_2_3_1_2"/><b>then </b>
<font color="Maroon" title="c13">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">b2</font> <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="pencil_3.html#E2:23_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="pencil_3.html#E6:23_1_2"><i><font color="Green" title="E31">A31</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="E13:23_1_2_3_1_2"/>
<font color="Maroon" title="c5">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> <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="pencil_3.html#E4:23_1_2_3_1_2"><i><font color="Green" title="E35">A41</font></i></a>, <a class="txt" href="pencil_3.html#E7:23_1_2_3_1_2"><i><font color="Green" title="E36">A42</font></i></a>, <a class="txt" href="pencil_3.html#E10:23_1_2_3_1_2"><i><font color="Green" title="E37">A43</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>let </b><font color="Maroon" title="c15">c1</font>, <font color="Maroon" title="c16">c2</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> <font color="Maroon" title="c14">r</font></span>)</span>;<br/>

<b>assume </b><a NAME="E14:23_1_2"/><i><font color="Green" title="E33">A44</font></i>: 
( <font color="Maroon" title="c5">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c15">c1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c6">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">c2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 ;<br/>

<a NAME="E15:23_1_2"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">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="E16:23_1_2"/><b>then </b>
<font color="Maroon" title="c11">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E4:23_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="pencil_3.html#E7:23_1"><i><font color="Green" title="E27">A6</font></i></a>, <a class="txt" href="pencil_3.html#E2:23_1_2"><i><font color="Green" title="E30">A30</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E17:23_1_2"/><b>then </b><i><font color="Green" title="E34">A45</font></i>: 
<font color="Maroon" title="c15">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E14:23_1_2"><i><font color="Green" title="E33">A44</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E18:23_1_2"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">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="E19:23_1_2"/><b>then </b>
<font color="Maroon" title="c13">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E6:23_1"><i><font color="Green" title="E26">A5</font></i></a>, <a class="txt" href="pencil_3.html#E8:23_1"><i><font color="Green" title="E28">A7</font></i></a>, <a class="txt" href="pencil_3.html#E6:23_1_2"><i><font color="Green" title="E31">A31</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E20:23_1_2"/><b>then </b>
<font color="Maroon" title="c16">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E14:23_1_2"><i><font color="Green" title="E33">A44</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="E21:23_1_2"/>
<font color="Maroon" title="c15">c1</font>,<font color="Maroon" title="c16">c2</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a> 
 <b>by </b><i><a class="txt" href="pencil_3.html#E10:23_1_2"><i><font color="Green" title="E32">A32</font></i></a>, <a class="txt" href="pencil_3.html#E17:23_1_2"><i><font color="Green" title="E34">A45</font></i></a></i>;<br/>


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


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

<b>assume </b><a NAME="E3:23"/><i><font color="Green" title="E23">A46</font></i>: 
for <font color="Olive" title="b1">b1</font>, <font color="Olive" title="b2">b2</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 <font color="Maroon" title="c3">B1</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">b1</font> &amp; <font color="Maroon" title="c4">B2</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">b2</font> holds <br/>(  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Olive" title="b1">b1</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="Olive" title="b2">b2</font> &amp;  ex <font color="Olive" title="b3">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> st <br/>( <font color="Olive" title="b3">r</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="Olive" title="b1">b1</font> &amp; ( for <font color="Olive" title="b4">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="b4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">r</font> holds <br/><font color="Olive" title="b1">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> ) &amp; ( for <font color="Olive" title="b4">c1</font>, <font color="Olive" title="b5">c2</font> being  <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> <font color="Olive" title="b3">r</font></span>)</span>  st <font color="Olive" title="b1">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b4">c1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Olive" title="b2">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b5">c2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds <br/><font color="Olive" title="b4">c1</font>,<font color="Olive" title="b5">c2</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a>  ) ) )
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">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="E5:23"/><i><font color="Green" title="E24">A47</font></i>: 
( <font color="Maroon" title="c3">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c5">L1</font> &amp; <font color="Maroon" title="c5">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="c5">L1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">L1</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_2.html#D2" title="PENCIL_2:def.2">PENCIL_2:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">L2</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="E7:23"/><i><font color="Green" title="E25">A48</font></i>: 
( <font color="Maroon" title="c4">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c6">L2</font> &amp; <font color="Maroon" title="c6">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="c6">L2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">L2</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_2.html#D2" title="PENCIL_2:def.2">PENCIL_2:def 2</a></i>;<br/>
<a NAME="E8:23"/><i><font color="Green" title="E26">A49</font></i>: 
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">L1</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="c6">L2</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E3:23"><i><font color="Green" title="E23">A46</font></i></a>, <a class="txt" href="pencil_3.html#E5:23"><i><font color="Green" title="E24">A47</font></i></a>, <a class="txt" href="pencil_3.html#E7:23"><i><font color="Green" title="E25">A48</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">r</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="E10:23"/><i><font color="Green" title="E27">A50</font></i>: 
( <font color="Maroon" title="c7">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">L1</font> &amp; ( 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> <font color="Maroon" title="c7">r</font> holds <br/><font color="Maroon" title="c5">L1</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="c6">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) &amp; ( for <font color="Olive" title="b1">c1</font>, <font color="Olive" title="b2">c2</font> being  <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> <font color="Maroon" title="c7">r</font></span>)</span>  st <font color="Maroon" title="c5">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">c1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c6">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b2">c2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds <br/><font color="Olive" title="b1">c1</font>,<font color="Olive" title="b2">c2</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a>  ) )
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:23"><i><font color="Green" title="E23">A46</font></i></a>, <a class="txt" href="pencil_3.html#E5:23"><i><font color="Green" title="E24">A47</font></i></a>, <a class="txt" href="pencil_3.html#E7:23"><i><font color="Green" title="E25">A48</font></i></a></i>;<br/>
<a NAME="E11:23"/>
( not <font color="Maroon" title="c5">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">r</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c5">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">r</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> )
 
<b>by </b><i><a class="txt" href="pencil_3.html#E10:23"><i><font color="Green" title="E27">A50</font></i></a>, <a class="ref" href="pencil_1.html#T12" title="PENCIL_1:th.12">PENCIL_1:12</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">c1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:23"/><i><font color="Green" title="E28">A51</font></i>: 
<font color="Maroon" title="c5">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">r</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="c8">c1</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="E14:23"/>
<font color="Maroon" title="c5">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>
 
<b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E15:23"/><b>then </b>
<font color="Maroon" title="c5">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">r</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="c7">r</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E16:23"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">c1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <font color="Maroon" title="c7">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E13:23"><i><font color="Green" title="E28">A51</font></i></a>, <a class="ref" href="pencil_3.html#T7" target="_self" title="PENCIL_3:th.7">Th7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">c1</font> = <font color="Maroon" title="c8">c1</font> 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> <font color="Maroon" title="c7">r</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/>
<a NAME="E18:23"/>
( not <font color="Maroon" title="c6">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">r</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c6">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">r</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> )
 
<b>by </b><i><a class="txt" href="pencil_3.html#E8:23"><i><font color="Green" title="E26">A49</font></i></a>, <a class="txt" href="pencil_3.html#E10:23"><i><font color="Green" title="E27">A50</font></i></a>, <a class="ref" href="pencil_1.html#T12" title="PENCIL_1:th.12">PENCIL_1:12</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">c2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E20:23"/><i><font color="Green" title="E29">A52</font></i>: 
<font color="Maroon" title="c6">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">r</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="c10">c2</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="E21:23"/>
<font color="Maroon" title="c6">L2</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="E22:23"/><b>then </b>
<font color="Maroon" title="c6">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">r</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="c7">r</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E23:23"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">c2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <font color="Maroon" title="c7">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E20:23"><i><font color="Green" title="E29">A52</font></i></a>, <a class="ref" href="pencil_3.html#T7" target="_self" title="PENCIL_3:th.7">Th7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">c2</font> = <font color="Maroon" title="c10">c2</font> 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> <font color="Maroon" title="c7">r</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/>
<div><i><font color="Green" title="E30">A53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_2"/><i><font color="Green" title="E31">A54</font></i>: 
<font color="Maroon" title="c9">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c2</font>
 ;<br/><i><font color="Green" title="E32">A55</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">L1</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#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">L2</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="23_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:23_2_2"/>
<font color="Maroon" title="c12">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">L1</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c13">s1</font> = <font color="Maroon" title="c12">s</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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="23_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_2_2_1"/>
( <font color="Maroon" title="c13">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> or <font color="Maroon" title="c13">s1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">r</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_2_2_1_1"><b>suppose </b></a><a NAME="E1:23_2_2_1_1"/>
<font color="Maroon" title="c13">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:23_2_2_1_1"/>
<font color="Maroon" title="c5">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E13:23"><i><font color="Green" title="E28">A51</font></i></a>, <a class="txt" href="pencil_3.html#E20:23"><i><font color="Green" title="E29">A52</font></i></a>, <a class="txt" href="pencil_3.html#E1:23_2"><i><font color="Green" title="E31">A54</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_2_2_1_2"><b>suppose </b></a><a NAME="E1:23_2_2_1_2"/>
<font color="Maroon" title="c13">s1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">r</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:23_2_2_1_2"/>
<font color="Maroon" title="c5">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E10:23"><i><font color="Green" title="E27">A50</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E4:23_2"/><b>then </b>
<font color="Maroon" title="c5">L1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c6">L2</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:23_2"><i><font color="Green" title="E32">A55</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/><a NAME="E5:23_2"/><b>then </b>
<font color="Maroon" title="c3">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:23"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="pencil_3.html#E5:23"><i><font color="Green" title="E24">A47</font></i></a>, <a class="txt" href="pencil_3.html#E7:23"><i><font color="Green" title="E25">A48</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E6:23_2"/>
contradiction
 <b>by </b><i><a class="txt" href="pencil_3.html#E5:23"><i><font color="Green" title="E24">A47</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E26:23"/>
<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c11">c2</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a> 
 
<b>by </b><i><a class="txt" href="pencil_3.html#E10:23"><i><font color="Green" title="E27">A50</font></i></a>, <a class="txt" href="pencil_3.html#E13:23"><i><font color="Green" title="E28">A51</font></i></a>, <a class="txt" href="pencil_3.html#E20:23"><i><font color="Green" title="E29">A52</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">bb</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> <font color="Maroon" title="c7">r</font></span>)</span><b> such that </b><br/><a NAME="E28:23"/><i><font color="Green" title="E31">A56</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c11">c2</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="c12">bb</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E25:23"><i><font color="Green" title="E30">A53</font></i></a>, <a class="ref" href="pencil_1.html#D1" title="PENCIL_1:def.1">PENCIL_1:def 1</a></i>;<br/>
<b>let </b><font color="Maroon" title="c13">x</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_3.html#D2" target="_self" title="PENCIL_3:def.2">PENCIL_3:def 2</a><br/>

<b>assume </b><a NAME="E29:23"/><i><font color="Green" title="E32">A57</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c14">x1</font> = <font color="Maroon" title="c13">x</font> as    <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</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_3.html#T6" target="_self" title="PENCIL_3:th.6">Th6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">x2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E32:23"/><i><font color="Green" title="E33">A58</font></i>: 
( <font color="Maroon" title="c15">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c15">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">L1</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>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">L1</font> holds <br/><font color="Maroon" title="c15">x2</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="c5">L1</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="pencil_3.html#E5:23"><i><font color="Green" title="E24">A47</font></i></a>, <a class="txt" href="pencil_3.html#E29:23"><i><font color="Green" title="E32">A57</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="c16">y</font> = <font color="Maroon" title="c14">x1</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c11">c2</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" title="PENCIL_1:th.25">PENCIL_1:25</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c16">y</font>
;<br/>

<b>reconsider </b><font color="Maroon" title="c17">y1</font> = <font color="Maroon" title="c16">y</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="E34">A59</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c17">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#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">L2</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="23_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:23_4"/>
<font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">L2</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c19">a1</font> = <font color="Maroon" title="c18">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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="23_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_4_1"/>
( <font color="Maroon" title="c19">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> or <font color="Maroon" title="c19">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">r</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_4_1_1"><b>suppose </b></a><a NAME="E1:23_4_1_1"/><i><font color="Green" title="E35">A60</font></i>: 
<font color="Maroon" title="c19">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font>
 ;<br/><div class="add"><a NAME="E2:23_4_1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c14">x1</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="E3:23_4_1_1"/><b>then </b>
<font color="Maroon" title="c17">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c2</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:23_4_1_1"><i><font color="Green" title="E35">A60</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="E4:23_4_1_1"/>
<font color="Maroon" title="c17">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E20:23"><i><font color="Green" title="E29">A52</font></i></a>, <a class="txt" href="pencil_3.html#E1:23_4_1_1"><i><font color="Green" title="E35">A60</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_4_1_2"><b>suppose </b></a><a NAME="E1:23_4_1_2"/><i><font color="Green" title="E35">A61</font></i>: 
<font color="Maroon" title="c19">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">r</font>
 ;<br/><div class="add"><a NAME="E2:23_4_1_2"/><i><font color="Green" title="E36">A62</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c14">x1</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="E3:23_4_1_2"/>
<font color="Maroon" title="c17">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">a1</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:23_4_1_2"><i><font color="Green" title="E35">A61</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><a NAME="E4:23_4_1_2"/><b>then </b>
<font color="Maroon" title="c17">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">a1</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E32:23"><i><font color="Green" title="E33">A58</font></i></a>, <a class="txt" href="pencil_3.html#E2:23_4_1_2"><i><font color="Green" title="E36">A62</font></i></a></i>;<br/><b>hence </b><a NAME="E5:23_4_1_2"/>
<font color="Maroon" title="c17">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E10:23"><i><font color="Green" title="E27">A50</font></i></a>, <a class="txt" href="pencil_3.html#E1:23_4_1_2"><i><font color="Green" title="E35">A61</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E37:23"/>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B2</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E7:23"><i><font color="Green" title="E25">A48</font></i></a>, <a class="txt" href="pencil_3.html#E35:23"><i><font color="Green" title="E34">A59</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="c18">B</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="c14">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c12">bb</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="ref" href="pencil_3.html#T16" target="_self" title="PENCIL_3:th.16">Th16</a></i>;<br/>
<i><font color="Green" title="E35">A63</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c14">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#K1" title="RELAT_1:func.1">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="c14">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c12">bb</font></span>)</span>
<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="23_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:23_6"/><i><font color="Green" title="E36">A64</font></i>: 
<font color="Maroon" title="c19">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c14">x1</font>
 ;<br/><a NAME="E2:23_6"/><b>then </b><i><font color="Green" title="E37">A65</font></i>: 
<font color="Maroon" title="c19">s</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="E3:23_6"/><b>then </b><i><font color="Green" title="E38">A66</font></i>: 
<font color="Maroon" title="c19">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c14">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span>
 <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="23_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_6_1"/>
( <font color="Maroon" title="c19">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> or <font color="Maroon" title="c19">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">r</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_6_1_1"><b>suppose </b></a><a NAME="E1:23_6_1_1"/><i><font color="Green" title="E39">A67</font></i>: 
<font color="Maroon" title="c19">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font>
 ;<br/><div class="add"><a NAME="E2:23_6_1_1"/>
<font color="Maroon" title="c14">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E32:23"><i><font color="Green" title="E33">A58</font></i></a>, <a class="txt" href="pencil_3.html#E1:23_6"><i><font color="Green" title="E36">A64</font></i></a></i>;<br/><a NAME="E3:23_6_1_1"/><b>then </b><i><font color="Green" title="E40">A68</font></i>: 
<font color="Maroon" title="c14">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">c1</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E13:23"><i><font color="Green" title="E28">A51</font></i></a>, <a class="txt" href="pencil_3.html#E1:23_6_1_1"><i><font color="Green" title="E39">A67</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E4:23_6_1_1"/>
<font color="Maroon" title="c12">bb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c14">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c12">bb</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:23_6"><i><font color="Green" title="E38">A66</font></i></a>, <a class="txt" href="pencil_3.html#E1:23_6_1_1"><i><font color="Green" title="E39">A67</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="E5:23_6_1_1"/>
<font color="Maroon" title="c14">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</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="c14">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c12">bb</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E28:23"><i><font color="Green" title="E31">A56</font></i></a>, <a class="txt" href="pencil_3.html#E3:23_6_1_1"><i><font color="Green" title="E40">A68</font></i></a>, <a class="ref" href="zfmisc_1.html#T38" title="ZFMISC_1:th.38">ZFMISC_1:38</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_6_1_2"><b>suppose </b></a><a NAME="E1:23_6_1_2"/>
<font color="Maroon" title="c19">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">r</font>
 ;<br/><div class="add"><a NAME="E2:23_6_1_2"/><b>then </b>
<span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c14">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="c19">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c14">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c12">bb</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</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="E3:23_6_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="c14">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c14">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c12">bb</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:23_6"><i><font color="Green" title="E37">A65</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="E4:23_6_1_2"/>
<font color="Maroon" title="c14">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</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="c14">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c12">bb</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E41:23"/><b>then </b><i><font color="Green" title="E36">A69</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">B</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E39:23"><i><font color="Green" title="E35">A63</font></i></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>
<i><font color="Green" title="E37">A70</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c17">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#K1" title="RELAT_1:func.1">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="c14">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c12">bb</font></span>)</span>
<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="23_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:23_8"/>
<font color="Maroon" title="c19">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c17">y1</font>
 ;<br/><a NAME="E2:23_8"/><b>then </b><i><font color="Green" title="E38">A71</font></i>: 
<font color="Maroon" title="c19">s</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="E3:23_8"/><b>then </b><i><font color="Green" title="E39">A72</font></i>: 
( <font color="Maroon" title="c19">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c14">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> &amp; <font color="Maroon" title="c19">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c14">x1</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="23_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_8_1"/>
( <font color="Maroon" title="c19">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> or <font color="Maroon" title="c19">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">r</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_8_1_1"><b>suppose </b></a><a NAME="E1:23_8_1_1"/><i><font color="Green" title="E40">A73</font></i>: 
<font color="Maroon" title="c19">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font>
 ;<br/><div class="add"><a NAME="E2:23_8_1_1"/><b>then </b><i><font color="Green" title="E41">A74</font></i>: 
<font color="Maroon" title="c17">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c2</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:23_8"><i><font color="Green" title="E39">A72</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><a NAME="E3:23_8_1_1"/>
<font color="Maroon" title="c12">bb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c14">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c12">bb</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:23_8"><i><font color="Green" title="E39">A72</font></i></a>, <a class="txt" href="pencil_3.html#E1:23_8_1_1"><i><font color="Green" title="E40">A73</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="E4:23_8_1_1"/>
<font color="Maroon" title="c17">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</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="c14">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c12">bb</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E28:23"><i><font color="Green" title="E31">A56</font></i></a>, <a class="txt" href="pencil_3.html#E2:23_8_1_1"><i><font color="Green" title="E41">A74</font></i></a>, <a class="ref" href="zfmisc_1.html#T38" title="ZFMISC_1:th.38">ZFMISC_1:38</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_8_1_2"><b>suppose </b></a><a NAME="E1:23_8_1_2"/><i><font color="Green" title="E40">A75</font></i>: 
<font color="Maroon" title="c19">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">r</font>
 ;<br/><div class="add"><a NAME="E2:23_8_1_2"/><b>then </b><i><font color="Green" title="E41">A76</font></i>: 
<font color="Maroon" title="c17">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</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="E3:23_8_1_2"/>
<span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c14">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="c19">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c14">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c12">bb</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:23_8_1_2"><i><font color="Green" title="E40">A75</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><a NAME="E4:23_8_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="c14">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c14">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c12">bb</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:23_8"><i><font color="Green" title="E38">A71</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="E5:23_8_1_2"/>
<font color="Maroon" title="c17">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</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="c14">x1</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c12">bb</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:23_8_1_2"><i><font color="Green" title="E41">A76</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E44:23"/><b>then </b>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">B</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E42:23"><i><font color="Green" title="E37">A70</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="E45:23"/><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="c13">x</font>,<font color="Maroon" title="c16">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="c18">B</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E41:23"><i><font color="Green" title="E36">A69</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="E46:23"/>
<font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c16">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" title="PENCIL_1:def.1">PENCIL_1:def 1</a></i>;<br/>


</div>
