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<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="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being <a href="pencil_1.html#V15" title="PENCIL_1:attr.15">PLS-yielding</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>  st ( for <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> holds  <font color="Olive" title="b1">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <font color="Olive" title="b2">i</font> is <a href="pencil_1.html#V10" title="PENCIL_1:attr.10">strongly_connected</a> ) holds <br/>for <font color="Olive" title="b2">f</font> being  <a href="pencil_2.html#NM2" title="PENCIL_2:NM.2">Collineation</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Olive" title="b1">A</font></span>)</span><br/> for <font color="Olive" title="b3">B1</font>, <font color="Olive" title="b4">B2</font> being   <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Olive" title="b1">A</font>  st <font color="Olive" title="b3">B1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b4">B2</font> &amp; <font color="Olive" title="b3">B1</font> <a href="pencil_3.html#R1" title="PENCIL_3:pred.1">'||'</a> <font color="Olive" title="b4">B2</font> holds <br/>for <font color="Olive" title="b5">b1</font>, <font color="Olive" title="b6">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="Olive" title="b1">A</font>  st  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b5">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">B1</font> &amp;  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">B2</font> holds <br/> <a href="pencil_3.html#K3" title="PENCIL_3:func.3">canonical_embedding</a> <font color="Olive" title="b2">f</font>,<font color="Olive" title="b5">b1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="pencil_3.html#K3" title="PENCIL_3:func.3">canonical_embedding</a> <font color="Olive" title="b2">f</font>,<font color="Olive" title="b6">b2</font></span><br/><b>let </b><font color="Maroon" title="c2">A</font> be  <a href="pencil_1.html#V15" title="PENCIL_1:attr.15">PLS-yielding</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> holds  <font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <font color="Olive" title="b1">i</font> is <a href="pencil_1.html#V10" title="PENCIL_1:attr.10">strongly_connected</a> ) implies for <font color="Olive" title="b1">f</font> being  <a href="pencil_2.html#NM2" title="PENCIL_2:NM.2">Collineation</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span><br/> for <font color="Olive" title="b2">B1</font>, <font color="Olive" title="b3">B2</font> being   <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b2">B1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b3">B2</font> &amp; <font color="Olive" title="b2">B1</font> <a href="pencil_3.html#R1" title="PENCIL_3:pred.1">'||'</a> <font color="Olive" title="b3">B2</font> holds <br/>for <font color="Olive" title="b4">b1</font>, <font color="Olive" title="b5">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  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B1</font> &amp;  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b5">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">B2</font> holds <br/> <a href="pencil_3.html#K3" title="PENCIL_3:func.3">canonical_embedding</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b4">b1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="pencil_3.html#K3" title="PENCIL_3:func.3">canonical_embedding</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b5">b2</font> )</span><br/>



<b>assume </b><a NAME="E1:30"/><i><font color="Green" title="E29">A1</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> holds  <font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <font color="Olive" title="b1">i</font> is <a href="pencil_1.html#V10" title="PENCIL_1:attr.10">strongly_connected</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="pencil_2.html#NM2" title="PENCIL_2:NM.2">Collineation</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span><br/> for <font color="Olive" title="b2">B1</font>, <font color="Olive" title="b3">B2</font> being   <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b2">B1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b3">B2</font> &amp; <font color="Olive" title="b2">B1</font> <a href="pencil_3.html#R1" title="PENCIL_3:pred.1">'||'</a> <font color="Olive" title="b3">B2</font> holds <br/>for <font color="Olive" title="b4">b1</font>, <font color="Olive" title="b5">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  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B1</font> &amp;  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b5">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">B2</font> holds <br/> <a href="pencil_3.html#K3" title="PENCIL_3:func.3">canonical_embedding</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b4">b1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="pencil_3.html#K3" title="PENCIL_3:func.3">canonical_embedding</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b5">b2</font></span><br/>

<b>let </b><font color="Maroon" title="c3">f</font> be   <a href="pencil_2.html#NM2" title="PENCIL_2:NM.2">Collineation</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">B1</font>, <font color="Olive" title="b2">B2</font> being   <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b1">B1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">B2</font> &amp; <font color="Olive" title="b1">B1</font> <a href="pencil_3.html#R1" title="PENCIL_3:pred.1">'||'</a> <font color="Olive" title="b2">B2</font> holds <br/>for <font color="Olive" title="b3">b1</font>, <font color="Olive" title="b4">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  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b3">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B1</font> &amp;  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b4">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B2</font> holds <br/> <a href="pencil_3.html#K3" title="PENCIL_3:func.3">canonical_embedding</a> <font color="Maroon" title="c3">f</font>,<font color="Olive" title="b3">b1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="pencil_3.html#K3" title="PENCIL_3:func.3">canonical_embedding</a> <font color="Maroon" title="c3">f</font>,<font color="Olive" title="b4">b2</font></span><br/>

<a NAME="E2:30"/><i><font color="Green" title="E30">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>let </b><font color="Maroon" title="c4">B1</font>, <font color="Maroon" title="c5">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">B1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">B2</font> &amp; <font color="Maroon" title="c4">B1</font> <a href="pencil_3.html#R1" title="PENCIL_3:pred.1">'||'</a> <font color="Maroon" title="c5">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  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b1">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B1</font> &amp;  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b2">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B2</font> holds <br/> <a href="pencil_3.html#K3" title="PENCIL_3:func.3">canonical_embedding</a> <font color="Maroon" title="c3">f</font>,<font color="Olive" title="b1">b1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="pencil_3.html#K3" title="PENCIL_3:func.3">canonical_embedding</a> <font color="Maroon" title="c3">f</font>,<font color="Olive" title="b2">b2</font> )</span><br/>

<b>assume </b><a NAME="E3:30"/><i><font color="Green" title="E31">A3</font></i>: 
( <font color="Maroon" title="c4">B1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">B2</font> &amp; <font color="Maroon" title="c4">B1</font> <a href="pencil_3.html#R1" title="PENCIL_3:pred.1">'||'</a> <font color="Maroon" title="c5">B2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">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  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b1">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B1</font> &amp;  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b2">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B2</font> holds <br/> <a href="pencil_3.html#K3" title="PENCIL_3:func.3">canonical_embedding</a> <font color="Maroon" title="c3">f</font>,<font color="Olive" title="b1">b1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="pencil_3.html#K3" title="PENCIL_3:func.3">canonical_embedding</a> <font color="Maroon" title="c3">f</font>,<font color="Olive" title="b2">b2</font></span><br/>

<b>let </b><font color="Maroon" title="c6">b1</font>, <font color="Maroon" title="c7">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B1</font> &amp;  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c7">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B2</font> implies  <a href="pencil_3.html#K3" title="PENCIL_3:func.3">canonical_embedding</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c6">b1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="pencil_3.html#K3" title="PENCIL_3:func.3">canonical_embedding</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c7">b2</font> )</span><br/>

<b>assume </b><a NAME="E4:30"/><i><font color="Green" title="E32">A4</font></i>: 
(  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B1</font> &amp;  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c7">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="pencil_3.html#K3" title="PENCIL_3:func.3">canonical_embedding</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c6">b1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="pencil_3.html#K3" title="PENCIL_3:func.3">canonical_embedding</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c7">b2</font></span><br/>

<a NAME="E5:30"/><i><font color="Green" title="E33">A5</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">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="E6:30"/><i><font color="Green" title="E34">A6</font></i>: 
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">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="c7">b2</font>
 
<b>by </b><i><a class="txt" href="#E3:30"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="#E4:30"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="pencil_3.html#T21" target="_self" title="PENCIL_3:th.21">Th21</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">i</font> =  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font>;<br/>
<b>consider </b><font color="Maroon" title="c9">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="E8:30"/><i><font color="Green" title="E35">A7</font></i>: 
( <font color="Maroon" title="c9">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="c6">b1</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="c9">r</font> holds <br/><font color="Maroon" title="c6">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="c7">b2</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="c9">r</font></span>)</span>  st <font color="Maroon" title="c6">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">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="c7">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">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="#E3:30"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="#E4:30"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="pencil_3.html#T21" target="_self" title="PENCIL_3:th.21">Th21</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">B3</font> = <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">B1</font>, <font color="Maroon" title="c11">B4</font> = <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">B2</font> as    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="txt" href="#E1:30"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="pencil_2.html#T24" title="PENCIL_2:th.24">PENCIL_2:24</a></i>;<br/>
<a NAME="E10:30"/>
<font color="Maroon" title="c3">f</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>
 
<b>by </b><i><a class="ref" href="pencil_2.html#D4" title="PENCIL_2:def.4">PENCIL_2:def 4</a></i>;<br/>
<a NAME="E11:30"/><b>then </b><i><font color="Green" title="E36">A8</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
;<br/>
<a NAME="E12:30"/><b>then </b><i><font color="Green" title="E37">A9</font></i>: 
( <font color="Maroon" title="c10">B3</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c11">B4</font> &amp; <font color="Maroon" title="c10">B3</font> <a href="pencil_3.html#R1" title="PENCIL_3:pred.1">'||'</a> <font color="Maroon" title="c11">B4</font> )
 
<b>by </b><i><a class="txt" href="#E3:30"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="pencil_3.html#T20" target="_self" title="PENCIL_3:th.20">Th20</a>, <a class="ref" href="funct_1.html#T125" title="FUNCT_1:th.125">FUNCT_1:125</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">b3</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="E14:30"/><i><font color="Green" title="E38">A10</font></i>: 
(  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c12">b3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B3</font> &amp; <font color="Maroon" title="c12">b3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c12">b3</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="c12">b3</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="c13">b4</font> being  non <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a> <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>  <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E16:30"/><i><font color="Green" title="E39">A11</font></i>: 
(  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c13">b4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">B4</font> &amp; <font color="Maroon" title="c13">b4</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="c13">b4</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="c13">b4</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="E17:30"/><i><font color="Green" title="E40">A12</font></i>: 
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c12">b3</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="c13">b4</font>
 
<b>by </b><i><a class="txt" href="#E12:30"><i><font color="Green" title="E37">A9</font></i></a>, <a class="txt" href="#E14:30"><i><font color="Green" title="E38">A10</font></i></a>, <a class="txt" href="#E16:30"><i><font color="Green" title="E39">A11</font></i></a>, <a class="ref" href="pencil_3.html#T21" target="_self" title="PENCIL_3:th.21">Th21</a></i>;<br/>
<b>set </b><font color="Maroon" title="c14">j</font> =  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c12">b3</font>;<br/>
<a NAME="E18:30"/><i><font color="Green" title="E41">A13</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pencil_3.html#K3" title="PENCIL_3:func.3">canonical_embedding</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c6">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><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">b1</font></span>)</span></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pencil_3.html#K3" title="PENCIL_3:func.3">canonical_embedding</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><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">b1</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E6:30"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">o</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="c6">b1</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><a href="pencil_3.html#K3" title="PENCIL_3:func.3">canonical_embedding</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pencil_3.html#K3" title="PENCIL_3:func.3">canonical_embedding</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><b>assume </b><a NAME="E1:30_1"/>
<font color="Maroon" title="c15">o</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="c6">b1</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="pencil_3.html#K3" title="PENCIL_3:func.3">canonical_embedding</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pencil_3.html#K3" title="PENCIL_3:func.3">canonical_embedding</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><b>then reconsider </b><font color="Maroon" title="c16">u</font> = <font color="Maroon" title="c15">o</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="c6">b1</font></span>)</span></span>)</span> ;<br/><b>consider </b><font color="Maroon" title="c17">p0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:30_1"/><i><font color="Green" title="E42">A14</font></i>: 
<font color="Maroon" title="c17">p0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c6">b1</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">p1</font> = <font color="Maroon" title="c17">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="#E4:30"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="#E4:30_1"><i><font color="Green" title="E42">A14</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">p</font> = <font color="Maroon" title="c18">p1</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="ref" href="pencil_1.html#T14" title="PENCIL_1:th.14">PENCIL_1:14</a></i>;<br/><b>set </b><font color="Maroon" title="c20">q</font> = <font color="Maroon" title="c19">p</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span>,<font color="Maroon" title="c16">u</font>;<br/><b>reconsider </b><font color="Maroon" title="c21">q1</font> = <font color="Maroon" title="c19">p</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span>,<font color="Maroon" title="c16">u</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/><a NAME="E8:30_1"/>
<font color="Maroon" title="c6">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">r</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>
 <b>by </b><i><a class="txt" href="#E8:30"><i><font color="Green" title="E35">A7</font></i></a>, <a class="ref" href="pencil_1.html#D21" title="PENCIL_1:def.21">PENCIL_1:def 21</a></i>;<br/><b>then consider </b><font color="Maroon" title="c22">c1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:30_1"/><i><font color="Green" title="E43">A15</font></i>: 
<font color="Maroon" title="c6">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">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="c22">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="E11:30_1"/>
<font color="Maroon" title="c6">b1</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="E12:30_1"/><b>then </b>
<font color="Maroon" title="c6">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">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="c9">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="E13:30_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c22">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="c9">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E10:30_1"><i><font color="Green" title="E43">A15</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="c23">c1</font> = <font color="Maroon" title="c22">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="c9">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="E15:30_1"/>
<font color="Maroon" title="c7">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">r</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>
 <b>by </b><i><a class="txt" href="#E6:30"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="#E8:30"><i><font color="Green" title="E35">A7</font></i></a>, <a class="ref" href="pencil_1.html#D21" title="PENCIL_1:def.21">PENCIL_1:def 21</a></i>;<br/><b>then consider </b><font color="Maroon" title="c24">c2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E17:30_1"/><i><font color="Green" title="E44">A16</font></i>: 
<font color="Maroon" title="c7">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">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="c24">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="E18:30_1"/>
<font color="Maroon" title="c7">b2</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="E19:30_1"/><b>then </b>
<font color="Maroon" title="c7">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">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="c9">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="E20:30_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c24">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="c9">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E17:30_1"><i><font color="Green" title="E44">A16</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="c25">c2</font> = <font color="Maroon" title="c24">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="c9">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/><b>set </b><font color="Maroon" title="c26">t</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span>,<font color="Maroon" title="c16">u</font></span>)</span> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c25">c2</font>;<br/><a NAME="E22:30_1"/>
<font color="Maroon" title="c19">p</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span>,<font color="Maroon" title="c16">u</font> is   <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>then reconsider </b><font color="Maroon" title="c27">t1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span>,<font color="Maroon" title="c16">u</font></span>)</span> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c25">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/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_1"/>
( <font color="Maroon" title="c23">c1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">c2</font> or <font color="Maroon" title="c23">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">c2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1"><b>suppose </b></a><a NAME="E1:30_1_1_1"/><i><font color="Green" title="E45">A17</font></i>: 
<font color="Maroon" title="c23">c1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">c2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="pencil_3.html#K3" title="PENCIL_3:func.3">canonical_embedding</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pencil_3.html#K3" title="PENCIL_3:func.3">canonical_embedding</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:30_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span>,<font color="Maroon" title="c16">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">r</font>
 <b>by </b><i><a class="txt" href="#E8:30"><i><font color="Green" title="E35">A7</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><a NAME="E3:30_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span>,<font color="Maroon" title="c16">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">r</font>
 <b>by </b><i><a class="txt" href="#E5:30"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="#E4:30_1"><i><font color="Green" title="E42">A14</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><a NAME="E4:30_1_1_1"/><b>then </b><i><font color="Green" title="E46">A18</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span>,<font color="Maroon" title="c16">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">c1</font>
 <b>by </b><i><a class="txt" href="#E10:30_1"><i><font color="Green" title="E43">A15</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E5:30_1_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span>,<font color="Maroon" title="c16">u</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E6:30_1_1_1"/><b>then </b><i><font color="Green" title="E47">A19</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span>,<font color="Maroon" title="c16">u</font></span>)</span> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c25">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">c2</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><a NAME="E7:30_1_1_1"/><b>then </b><i><font color="Green" title="E48">A20</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">t1</font>
 <b>by </b><i><a class="txt" href="#E2:30"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="#E11:30"><i><font color="Green" title="E36">A8</font></i></a>, <a class="txt" href="#E1:30_1_1_1"><i><font color="Green" title="E45">A17</font></i></a>, <a class="txt" href="#E4:30_1_1_1"><i><font color="Green" title="E46">A18</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c28">q3</font>, <font color="Maroon" title="c29">t3</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c28">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">q1</font> &amp; <font color="Maroon" title="c29">t3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">t1</font> implies  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/>( ( for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">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="Olive" title="b1">r</font></span>)</span>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">t3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> holds <br/>( <font color="Olive" title="b2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">b</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a>  ) ) &amp; ( for <font color="Olive" title="b2">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="b2">j</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="c28">q3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">t3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> ) ) )</span><br/><b>assume </b><a NAME="E1:30_1_1_1_1"/><i><font color="Green" title="E49">A21</font></i>: 
( <font color="Maroon" title="c28">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">q1</font> &amp; <font color="Maroon" title="c29">t3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">t1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">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/>( ( for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">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="Olive" title="b1">r</font></span>)</span>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">t3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> holds <br/>( <font color="Olive" title="b2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">b</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a>  ) ) &amp; ( for <font color="Olive" title="b2">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="b2">j</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="c28">q3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">t3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> ) )</span><br/><b>take </b><font color="Maroon" title="c30">r</font> = <font color="Maroon" title="c9">r</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">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="c30">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="c28">q3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">r</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">t3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">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="c30">r</font> holds <br/><font color="Maroon" title="c28">q3</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="c29">t3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> ) )</span><br/><b>thus </b><a NAME="E2:30_1_1_1_1"/>
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="c30">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="c28">q3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">r</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">t3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">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>  )
 <b>by </b><i><a class="txt" href="#E8:30"><i><font color="Green" title="E35">A7</font></i></a>, <a class="txt" href="#E10:30_1"><i><font color="Green" title="E43">A15</font></i></a>, <a class="txt" href="#E17:30_1"><i><font color="Green" title="E44">A16</font></i></a>, <a class="txt" href="#E1:30_1_1_1"><i><font color="Green" title="E45">A17</font></i></a>, <a class="txt" href="#E4:30_1_1_1"><i><font color="Green" title="E46">A18</font></i></a>, <a class="txt" href="#E6:30_1_1_1"><i><font color="Green" title="E47">A19</font></i></a>, <a class="txt" href="#E1:30_1_1_1_1"><i><font color="Green" title="E49">A21</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">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="c30">r</font> holds <br/><font color="Maroon" title="c28">q3</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="c29">t3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font></span><br/><b>let </b><font color="Maroon" title="c31">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c31">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c30">r</font> implies <font color="Maroon" title="c28">q3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">t3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">j</font> )</span><br/><b>assume </b><a NAME="E3:30_1_1_1_1"/>
<font color="Maroon" title="c31">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c30">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c28">q3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">t3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">j</font></span><br/><b>hence </b><a NAME="E4:30_1_1_1_1"/>
<font color="Maroon" title="c28">q3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">t3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">j</font>
 <b>by </b><i><a class="txt" href="#E1:30_1_1_1_1"><i><font color="Green" title="E49">A21</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E9:30_1_1_1"/><b>then </b>
<font color="Maroon" title="c21">q1</font>,<font color="Maroon" title="c27">t1</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a> 
 <b>by </b><i><a class="txt" href="#E1:30_1_1_1"><i><font color="Green" title="E45">A17</font></i></a>, <a class="txt" href="#E4:30_1_1_1"><i><font color="Green" title="E46">A18</font></i></a>, <a class="txt" href="#E6:30_1_1_1"><i><font color="Green" title="E47">A19</font></i></a>, <a class="ref" href="pencil_3.html#T17" target="_self" title="PENCIL_3:th.17">Th17</a></i>;<br/><a NAME="E10:30_1_1_1"/><b>then </b><i><font color="Green" title="E49">A22</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">q1</font>,<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">t1</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a> 
 <b>by </b><i><a class="ref" href="pencil_3.html#T1" target="_self" title="PENCIL_3:th.1">Th1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c28">fq</font> = <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">q1</font>, <font color="Maroon" title="c29">ft</font> = <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">t1</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="ref" href="pencil_1.html#T14" title="PENCIL_1:th.14">PENCIL_1:14</a></i>;<br/><a NAME="E12:30_1_1_1"/><i><font color="Green" title="E50">A23</font></i>: 
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c12">b3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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">b1</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:30"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="#E4:30"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="#E14:30"><i><font color="Green" title="E38">A10</font></i></a>, <a class="ref" href="pencil_3.html#D3" target="_self" title="PENCIL_3:def.3">Def3</a></i>;<br/><a NAME="E13:30_1_1_1"/><i><font color="Green" title="E51">A24</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span>,<font color="Maroon" title="c16">u</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c19">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span>,<font color="Maroon" title="c16">u</font></span>)</span> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c25">c2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">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/><div><i><font color="Green" title="E52">A25</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c30">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c30">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span>,<font color="Maroon" title="c16">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><b>assume </b><a NAME="E1:30_1_1_1_2"/><i><font color="Green" title="E53">A26</font></i>: 
<font color="Maroon" title="c30">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span>,<font color="Maroon" title="c16">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_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="E1:30_1_1_1_2_1"/>
( <font color="Maroon" title="c30">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font> or <font color="Maroon" title="c30">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:30_1_1_1_2_1_1"/>
<font color="Maroon" title="c30">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span>,<font color="Maroon" title="c16">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:30_1_1_1_2_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span>,<font color="Maroon" title="c16">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">u</font> &amp; <font color="Maroon" title="c6">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><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">b1</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E4:30"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="#E13:30_1_1_1"><i><font color="Green" title="E51">A24</font></i></a>, <a class="ref" href="pencil_3.html#T10" target="_self" title="PENCIL_3:th.10">Th10</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="E3:30_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span>,<font color="Maroon" title="c16">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:30_1_1_1_2_1_2"/>
<font color="Maroon" title="c30">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span>,<font color="Maroon" title="c16">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:30_1_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span>,<font color="Maroon" title="c16">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">a</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E3:30_1_1_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span>,<font color="Maroon" title="c16">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">a</font>
 <b>by </b><i><a class="txt" href="#E4:30_1"><i><font color="Green" title="E42">A14</font></i></a>, <a class="txt" href="#E13:30_1_1_1"><i><font color="Green" title="E51">A24</font></i></a>, <a class="txt" href="#E1:30_1_1_1_2"><i><font color="Green" title="E53">A26</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E15:30_1_1_1"/><b>then </b><i><font color="Green" title="E53">A27</font></i>: 
<font color="Maroon" title="c19">p</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span>,<font color="Maroon" title="c16">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c6">b1</font>
 <b>by </b><i><a class="txt" href="#E13:30_1_1_1"><i><font color="Green" title="E51">A24</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c30">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c30">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span>,<font color="Maroon" title="c16">u</font></span>)</span> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c25">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><b>assume </b><a NAME="E1:30_1_1_1_3"/><i><font color="Green" title="E54">A28</font></i>: 
<font color="Maroon" title="c30">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span>,<font color="Maroon" title="c16">u</font></span>)</span> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c25">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_1_1_3_1"/>
( <font color="Maroon" title="c30">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r</font> or <font color="Maroon" title="c30">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">r</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:30_1_1_1_3_1_1"/><i><font color="Green" title="E55">A29</font></i>: 
<font color="Maroon" title="c30">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span>,<font color="Maroon" title="c16">u</font></span>)</span> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c25">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:30_1_1_1_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span>,<font color="Maroon" title="c16">u</font></span>)</span> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c25">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">c2</font>
 <b>by </b><i><a class="txt" href="#E13:30_1_1_1"><i><font color="Green" title="E51">A24</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="E3:30_1_1_1_3_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span>,<font color="Maroon" title="c16">u</font></span>)</span> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c25">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">a</font>
 <b>by </b><i><a class="txt" href="#E17:30_1"><i><font color="Green" title="E44">A16</font></i></a>, <a class="txt" href="#E1:30_1_1_1_3_1_1"><i><font color="Green" title="E55">A29</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:30_1_1_1_3_1_2"/><i><font color="Green" title="E55">A30</font></i>: 
<font color="Maroon" title="c30">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span>,<font color="Maroon" title="c16">u</font></span>)</span> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c25">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:30_1_1_1_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span>,<font color="Maroon" title="c16">u</font></span>)</span> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c25">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span>,<font color="Maroon" title="c16">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">a</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="E3:30_1_1_1_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span>,<font color="Maroon" title="c16">u</font></span>)</span> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c25">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">a</font>
 <b>by </b><i><a class="txt" href="#E14:30_1_1_1"><i><font color="Green" title="E52">A25</font></i></a>, <a class="txt" href="#E1:30_1_1_1_3"><i><font color="Green" title="E54">A28</font></i></a></i>;<br/><b>hence </b><a NAME="E4:30_1_1_1_3_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span>,<font color="Maroon" title="c16">u</font></span>)</span> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c25">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">a</font>
 <b>by </b><i><a class="txt" href="#E8:30"><i><font color="Green" title="E35">A7</font></i></a>, <a class="txt" href="#E1:30_1_1_1_3"><i><font color="Green" title="E54">A28</font></i></a>, <a class="txt" href="#E1:30_1_1_1_3_1_2"><i><font color="Green" title="E55">A30</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E17:30_1_1_1"/><b>then </b><i><font color="Green" title="E54">A31</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span>,<font color="Maroon" title="c16">u</font></span>)</span> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c25">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c7">b2</font>
 <b>by </b><i><a class="txt" href="#E13:30_1_1_1"><i><font color="Green" title="E51">A24</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><a NAME="E18:30_1_1_1"/><b>then </b><i><font color="Green" title="E55">A32</font></i>: 
<span class="p1">(<span class="default"><a href="pencil_3.html#K3" title="PENCIL_3:func.3">canonical_embedding</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p4">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span>,<font color="Maroon" title="c16">u</font></span>)</span> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c25">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">ft</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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">b1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:30"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="#E4:30"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="#E6:30"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="pencil_3.html#D4" target="_self" title="PENCIL_3:def.4">Def4</a></i>;<br/><div><i><font color="Green" title="E56">A33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_1_1_1_4"/><i><font color="Green" title="E57">A34</font></i>: 
<font color="Maroon" title="c28">fq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c12">b3</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c29">ft</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c12">b3</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>consider </b><font color="Maroon" title="c30">l</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="E3:30_1_1_1_4"/><i><font color="Green" title="E58">A35</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="c30">l</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="c28">fq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">l</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">ft</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">l</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="c30">l</font> holds <br/><font color="Maroon" title="c28">fq</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="c29">ft</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="#E7:30_1_1_1"><i><font color="Green" title="E48">A20</font></i></a>, <a class="txt" href="#E10:30_1_1_1"><i><font color="Green" title="E49">A22</font></i></a>, <a class="ref" href="pencil_3.html#T17" target="_self" title="PENCIL_3:th.17">Th17</a></i>;<br/><a NAME="E4:30_1_1_1_4"/><i><font color="Green" title="E59">A36</font></i>: 
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c12">b3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">l</font>
 <b>by </b><i><a class="txt" href="#E1:30_1_1_1_4"><i><font color="Green" title="E57">A34</font></i></a>, <a class="txt" href="#E3:30_1_1_1_4"><i><font color="Green" title="E58">A35</font></i></a></i>;<br/><a NAME="E5:30_1_1_1_4"/><i><font color="Green" title="E60">A37</font></i>: 
<font color="Maroon" title="c28">fq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B3</font>
 <b>by </b><i><a class="txt" href="#E2:30"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="#E4:30"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="#E15:30_1_1_1"><i><font color="Green" title="E53">A27</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E6:30_1_1_1_4"/><i><font color="Green" title="E61">A38</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">fq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c29">ft</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">b3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">b4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> )
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c31">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c31">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> implies <font color="Maroon" title="c28">fq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">b4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><b>assume </b><a NAME="E1:30_1_1_1_4_1"/>
<font color="Maroon" title="c31">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c28">fq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">b4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><b>then reconsider </b><font color="Maroon" title="c32">o1</font> = <font color="Maroon" title="c31">o</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_1_1_4_1_1"/>
( <font color="Maroon" title="c32">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c12">b3</font> or <font color="Maroon" title="c32">o1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c12">b3</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:30_1_1_1_4_1_1_1"/>
<font color="Maroon" title="c32">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c12">b3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c28">fq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">b4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><b>hence </b><a NAME="E2:30_1_1_1_4_1_1_1"/>
<font color="Maroon" title="c28">fq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">b4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">o</font>
 <b>by </b><i><a class="txt" href="#E14:30"><i><font color="Green" title="E38">A10</font></i></a>, <a class="txt" href="#E16:30"><i><font color="Green" title="E39">A11</font></i></a>, <a class="txt" href="#E17:30"><i><font color="Green" title="E40">A12</font></i></a>, <a class="txt" href="#E5:30_1_1_1_4"><i><font color="Green" title="E60">A37</font></i></a>, <a class="txt" href="#E6:30_1_1_1_4"><i><font color="Green" title="E61">A38</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:30_1_1_1_4_1_1_2"/>
<font color="Maroon" title="c32">o1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c12">b3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c28">fq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">b4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:30_1_1_1_4_1_1_2"/><b>then </b><i><font color="Green" title="E62">A39</font></i>: 
<font color="Maroon" title="c28">fq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">ft</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">o1</font>
 <b>by </b><i><a class="txt" href="#E3:30_1_1_1_4"><i><font color="Green" title="E58">A35</font></i></a>, <a class="txt" href="#E4:30_1_1_1_4"><i><font color="Green" title="E59">A36</font></i></a></i>;<br/><a NAME="E3:30_1_1_1_4_1_1_2"/>
<font color="Maroon" title="c29">ft</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c13">b4</font>
 <b>by </b><i><a class="txt" href="#E2:30"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="#E4:30"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="#E16:30"><i><font color="Green" title="E39">A11</font></i></a>, <a class="txt" href="#E17:30_1_1_1"><i><font color="Green" title="E54">A31</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><b>hence </b><a NAME="E4:30_1_1_1_4_1_1_2"/>
<font color="Maroon" title="c28">fq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">b4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">o</font>
 <b>by </b><i><a class="txt" href="#E6:30_1_1_1_4"><i><font color="Green" title="E61">A38</font></i></a>, <a class="txt" href="#E2:30_1_1_1_4_1_1_2"><i><font color="Green" title="E62">A39</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E8:30_1_1_1_4"/><b>then </b>
<font color="Maroon" title="c28">fq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c13">b4</font>
 <b>by </b><i><a class="txt" href="#E6:30_1_1_1_4"><i><font color="Green" title="E61">A38</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><a NAME="E9:30_1_1_1_4"/><b>then </b>
<font color="Maroon" title="c28">fq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c12">b3</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c13">b4</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E14:30"><i><font color="Green" title="E38">A10</font></i></a>, <a class="txt" href="#E5:30_1_1_1_4"><i><font color="Green" title="E60">A37</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E10:30_1_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="#E12:30"><i><font color="Green" title="E37">A9</font></i></a>, <a class="txt" href="#E14:30"><i><font color="Green" title="E38">A10</font></i></a>, <a class="txt" href="#E16:30"><i><font color="Green" title="E39">A11</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E20:30_1_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c19">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="E21:30_1_1_1"/><b>then </b><i><font color="Green" title="E57">A40</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span>,<font color="Maroon" title="c16">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">o</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><a NAME="E22:30_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span>,<font color="Maroon" title="c16">u</font></span>)</span> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c25">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">o</font>
 <b>by </b><i><a class="txt" href="#E8:30"><i><font color="Green" title="E35">A7</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="E23:30_1_1_1"/>
<span class="p1">(<span class="default"><a href="pencil_3.html#K3" title="PENCIL_3:func.3">canonical_embedding</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pencil_3.html#K3" title="PENCIL_3:func.3">canonical_embedding</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <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="#E1:30"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="#E4:30"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="#E12:30_1_1_1"><i><font color="Green" title="E50">A23</font></i></a>, <a class="txt" href="#E15:30_1_1_1"><i><font color="Green" title="E53">A27</font></i></a>, <a class="txt" href="#E18:30_1_1_1"><i><font color="Green" title="E55">A32</font></i></a>, <a class="txt" href="#E19:30_1_1_1"><i><font color="Green" title="E56">A33</font></i></a>, <a class="txt" href="#E21:30_1_1_1"><i><font color="Green" title="E57">A40</font></i></a>, <a class="ref" href="pencil_3.html#D4" target="_self" title="PENCIL_3:def.4">Def4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_2"><b>suppose </b></a><a NAME="E1:30_1_1_2"/><i><font color="Green" title="E45">A41</font></i>: 
<font color="Maroon" title="c23">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">c2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="pencil_3.html#K3" title="PENCIL_3:func.3">canonical_embedding</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pencil_3.html#K3" title="PENCIL_3:func.3">canonical_embedding</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:30_1_1_2"/><i><font color="Green" title="E46">A42</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c28">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c28">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> implies <font color="Maroon" title="c6">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><b>assume </b><a NAME="E1:30_1_1_2_1"/>
<font color="Maroon" title="c28">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><b>then reconsider </b><font color="Maroon" title="c29">o1</font> = <font color="Maroon" title="c28">o</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_1_2_1_1"/>
( <font color="Maroon" title="c9">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">o1</font> or <font color="Maroon" title="c9">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c29">o1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:30_1_1_2_1_1_1"/>
<font color="Maroon" title="c9">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">o1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><b>hence </b><a NAME="E2:30_1_1_2_1_1_1"/>
<font color="Maroon" title="c6">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">o</font>
 <b>by </b><i><a class="txt" href="#E10:30_1"><i><font color="Green" title="E43">A15</font></i></a>, <a class="txt" href="#E17:30_1"><i><font color="Green" title="E44">A16</font></i></a>, <a class="txt" href="#E1:30_1_1_2"><i><font color="Green" title="E45">A41</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:30_1_1_2_1_1_2"/>
<font color="Maroon" title="c9">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c29">o1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><b>hence </b><a NAME="E2:30_1_1_2_1_1_2"/>
<font color="Maroon" title="c6">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">o</font>
 <b>by </b><i><a class="txt" href="#E8:30"><i><font color="Green" title="E35">A7</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:30_1_1_2"/>
<span class="p1">(<span class="default"><a href="pencil_3.html#K3" title="PENCIL_3:func.3">canonical_embedding</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pencil_3.html#K3" title="PENCIL_3:func.3">canonical_embedding</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <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="#E2:30_1_1_2"><i><font color="Green" title="E46">A42</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E20:30"/>
 <a href="pencil_3.html#K3" title="PENCIL_3:func.3">canonical_embedding</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c6">b1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="pencil_3.html#K3" title="PENCIL_3:func.3">canonical_embedding</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c7">b2</font>
 <b>by </b><i><a class="txt" href="#E18:30"><i><font color="Green" title="E41">A13</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
