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

<b>let </b><font color="Maroon" title="c1">I</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being <a href="waybel_3.html#V4" title="WAYBEL_3:attr.4">non-Empty</a> <a href="pencil_1.html#V11" title="PENCIL_1:attr.11">TopStruct-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 <br/> ( <font color="Olive" title="b1">A</font> <a href="pencil_1.html#K1" title="PENCIL_1:func.1">.</a> <font color="Olive" title="b2">i</font> is <a href="pencil_1.html#V6" title="PENCIL_1:attr.6">identifying_close_blocks</a> &amp; <font color="Olive" title="b1">A</font> <a href="pencil_1.html#K1" title="PENCIL_1:func.1">.</a> <font color="Olive" title="b2">i</font> is <a href="pencil_1.html#V5" title="PENCIL_1:attr.5">with_non_trivial_blocks</a> &amp; not for <font color="Olive" title="b3">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#K1" title="PENCIL_1:func.1">.</a> <font color="Olive" title="b3">i</font> is <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a> ) ) holds <br/> <a href="pencil_1.html#K5" title="PENCIL_1:func.5">Segre_Product</a> <font color="Olive" title="b1">A</font> is <a href="pencil_1.html#V6" title="PENCIL_1:attr.6">identifying_close_blocks</a></span><br/><b>let </b><font color="Maroon" title="c2">A</font> be  <a href="waybel_3.html#V4" title="WAYBEL_3:attr.4">non-Empty</a> <a href="pencil_1.html#V11" title="PENCIL_1:attr.11">TopStruct-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 <br/> ( <font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K1" title="PENCIL_1:func.1">.</a> <font color="Olive" title="b1">i</font> is <a href="pencil_1.html#V6" title="PENCIL_1:attr.6">identifying_close_blocks</a> &amp; <font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K1" title="PENCIL_1:func.1">.</a> <font color="Olive" title="b1">i</font> is <a href="pencil_1.html#V5" title="PENCIL_1:attr.5">with_non_trivial_blocks</a> &amp; not 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="Maroon" title="c2">A</font> <a href="pencil_1.html#K1" title="PENCIL_1:func.1">.</a> <font color="Olive" title="b2">i</font> is <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a> ) ) implies  <a href="pencil_1.html#K5" title="PENCIL_1:func.5">Segre_Product</a> <font color="Maroon" title="c2">A</font> is <a href="pencil_1.html#V6" title="PENCIL_1:attr.6">identifying_close_blocks</a> )</span><br/>



<b>assume </b><a NAME="E1:54"/><i><font color="Green" title="E37">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 <br/> ( <font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K1" title="PENCIL_1:func.1">.</a> <font color="Olive" title="b1">i</font> is <a href="pencil_1.html#V6" title="PENCIL_1:attr.6">identifying_close_blocks</a> &amp; <font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K1" title="PENCIL_1:func.1">.</a> <font color="Olive" title="b1">i</font> is <a href="pencil_1.html#V5" title="PENCIL_1:attr.5">with_non_trivial_blocks</a> &amp; not 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="Maroon" title="c2">A</font> <a href="pencil_1.html#K1" title="PENCIL_1:func.1">.</a> <font color="Olive" title="b2">i</font> is <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="pencil_1.html#K5" title="PENCIL_1:func.5">Segre_Product</a> <font color="Maroon" title="c2">A</font> is <a href="pencil_1.html#V6" title="PENCIL_1:attr.6">identifying_close_blocks</a></span><br/>

<a NAME="E2:54"/>
for <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">l</font> being  <a href="pencil_1.html#NM1" title="PENCIL_1:NM.1">Block</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K5" title="PENCIL_1:func.5">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span>  st 2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">l</font></span>)</span> holds <br/><font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">l</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">k</font>, <font color="Maroon" title="c4">l</font> be   <a href="pencil_1.html#NM1" title="PENCIL_1:NM.1">Block</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K5" title="PENCIL_1:func.5">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">l</font></span>)</span> implies <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l</font> )</span><br/>

<a NAME="E1:54_1"/>
not  <a href="pencil_1.html#K5" title="PENCIL_1:func.5">Segre_Product</a> <font color="Maroon" title="c2">A</font> is <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>
 
<b>by </b><i><a class="txt" href="#E1:54"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="pencil_1.html#T15" target="_self" title="PENCIL_1:th.15">Th15</a></i>;<br/>
<a NAME="E2:54_1"/><b>then </b>
not  <a href="pencil_1.html#K4" title="PENCIL_1:func.4">Segre_Blocks</a> <font color="Maroon" title="c2">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c5">B</font> being  <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>  <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E4:54_1"/><i><font color="Green" title="E38">A2</font></i>: 
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c5">B</font> &amp;  ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> st <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is   <a href="pencil_1.html#NM1" title="PENCIL_1:NM.1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K1" title="PENCIL_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_1.html#D22" target="_self" title="PENCIL_1:def.22">Def22</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">i</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="E6:54_1"/><i><font color="Green" title="E39">A3</font></i>: 
<font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> is   <a href="pencil_1.html#NM1" title="PENCIL_1:NM.1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K1" title="PENCIL_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:54_1"><i><font color="Green" title="E38">A2</font></i></a></i>;<br/>
<a NAME="E7:54_1"/><i><font color="Green" title="E40">A4</font></i>: 
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="c6">i</font> holds <br/>( not <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">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="c7">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">i</font> implies ( not <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> ) )</span><br/>

<b>assume </b><a NAME="E1:54_1_1"/><i><font color="Green" title="E41">A5</font></i>: 
<font color="Maroon" title="c7">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> )</span><br/>

<b>consider </b><font color="Maroon" title="c8">i1</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="E3:54_1_1"/><i><font color="Green" title="E42">A6</font></i>: 
for <font color="Olive" title="b1">j1</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">j1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">i1</font> holds <br/>( not <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j1</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> )
 <b>by </b><i><a class="ref" href="pencil_1.html#D20" target="_self" title="PENCIL_1:def.20">Def20</a></i>;<br/>
<a NAME="E4:54_1_1"/>
<font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K1" title="PENCIL_1:func.1">.</a> <font color="Maroon" title="c6">i</font> is <a href="pencil_1.html#V5" title="PENCIL_1:attr.5">with_non_trivial_blocks</a>
 
<b>by </b><i><a class="txt" href="#E1:54"><i><font color="Green" title="E37">A1</font></i></a></i>;<br/>
<a NAME="E5:54_1_1"/><b>then </b>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E6:54_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="pencil_1.html#D6" target="_self" title="PENCIL_1:def.6">Def6</a></i>;<br/>
<a NAME="E6:54_1_1"/><b>then </b>
not <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>
 
<b>by </b><i><a class="ref" href="pencil_1.html#T4" target="_self" title="PENCIL_1:th.4">Th4</a></i>;<br/>
<a NAME="E7:54_1_1"/><b>then </b>
<font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i</font>
 
<b>by </b><i><a class="txt" href="#E3:54_1_1"><i><font color="Green" title="E42">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:54_1_1"/>
( not <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> )
 <b>by </b><i><a class="txt" href="#E1:54_1_1"><i><font color="Green" title="E41">A5</font></i></a>, <a class="txt" href="#E3:54_1_1"><i><font color="Green" title="E42">A6</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>
<a NAME="E8:54_1"/>
not  <a href="pencil_1.html#K5" title="PENCIL_1:func.5">Segre_Product</a> <font color="Maroon" title="c2">A</font> is <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>
 
<b>by </b><i><a class="txt" href="#E1:54"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="pencil_1.html#T15" target="_self" title="PENCIL_1:th.15">Th15</a></i>;<br/>
<a NAME="E9:54_1"/><b>then </b>
not  <a href="pencil_1.html#K4" title="PENCIL_1:func.4">Segre_Blocks</a> <font color="Maroon" title="c2">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c7">C</font> being  <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>  <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E11:54_1"/><i><font color="Green" title="E41">A7</font></i>: 
( <font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c7">C</font> &amp;  ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> st <font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is   <a href="pencil_1.html#NM1" title="PENCIL_1:NM.1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K1" title="PENCIL_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_1.html#D22" target="_self" title="PENCIL_1:def.22">Def22</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font><b> such that </b><br/><a NAME="E13:54_1"/><i><font color="Green" title="E42">A8</font></i>: 
<font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font> is   <a href="pencil_1.html#NM1" title="PENCIL_1:NM.1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K1" title="PENCIL_1:func.1">.</a> <font color="Maroon" title="c8">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E11:54_1"><i><font color="Green" title="E41">A7</font></i></a></i>;<br/>
<a NAME="E14:54_1"/><i><font color="Green" title="E43">A9</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>  st <font color="Maroon" title="c8">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">i</font> holds <br/>( not <font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">i</font> implies ( not <font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> ) )</span><br/>

<b>assume </b><a NAME="E1:54_1_2"/><i><font color="Green" title="E44">A10</font></i>: 
<font color="Maroon" title="c8">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> )</span><br/>

<b>consider </b><font color="Maroon" title="c10">j1</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="E3:54_1_2"/><i><font color="Green" title="E45">A11</font></i>: 
for <font color="Olive" title="b1">i1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>  st <font color="Maroon" title="c10">j1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">i1</font> holds <br/>( not <font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i1</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> )
 <b>by </b><i><a class="ref" href="pencil_1.html#D20" target="_self" title="PENCIL_1:def.20">Def20</a></i>;<br/>
<a NAME="E4:54_1_2"/>
<font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K1" title="PENCIL_1:func.1">.</a> <font color="Maroon" title="c8">j</font> is <a href="pencil_1.html#V5" title="PENCIL_1:attr.5">with_non_trivial_blocks</a>
 
<b>by </b><i><a class="txt" href="#E1:54"><i><font color="Green" title="E37">A1</font></i></a></i>;<br/>
<a NAME="E5:54_1_2"/><b>then </b>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E13:54_1"><i><font color="Green" title="E42">A8</font></i></a>, <a class="ref" href="pencil_1.html#D6" target="_self" title="PENCIL_1:def.6">Def6</a></i>;<br/>
<a NAME="E6:54_1_2"/><b>then </b>
not <font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>
 
<b>by </b><i><a class="ref" href="pencil_1.html#T4" target="_self" title="PENCIL_1:th.4">Th4</a></i>;<br/>
<a NAME="E7:54_1_2"/><b>then </b>
<font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font>
 
<b>by </b><i><a class="txt" href="#E3:54_1_2"><i><font color="Green" title="E45">A11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:54_1_2"/>
( not <font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> )
 <b>by </b><i><a class="txt" href="#E1:54_1_2"><i><font color="Green" title="E44">A10</font></i></a>, <a class="txt" href="#E3:54_1_2"><i><font color="Green" title="E45">A11</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>
<b>assume </b><a NAME="E15:54_1"/>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">l</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l</font></span><br/>

<b>then consider </b><font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c10">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E17:54_1"/><i><font color="Green" title="E44">A12</font></i>: 
( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">k</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">l</font> &amp; <font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">k</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">l</font> &amp; <font color="Maroon" title="c9">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b</font> )
 <b>by </b><i><a class="ref" href="pencil_1.html#T2" target="_self" title="PENCIL_1:th.2">Th2</a></i>;<br/>
<a NAME="E18:54_1"/><i><font color="Green" title="E45">A13</font></i>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c5">B</font>
 
<b>by </b><i><a class="txt" href="#E4:54_1"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="#E17:54_1"><i><font color="Green" title="E44">A12</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">a</font> = <font color="Maroon" title="c9">a</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="ref" href="card_3.html#T64" title="CARD_3:th.64">CARD_3:64</a></i>;<br/>
<a NAME="E20:54_1"/><i><font color="Green" title="E46">A14</font></i>: 
<font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c5">B</font>
 
<b>by </b><i><a class="txt" href="#E4:54_1"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="#E17:54_1"><i><font color="Green" title="E44">A12</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">b</font> = <font color="Maroon" title="c10">b</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="ref" href="card_3.html#T64" title="CARD_3:th.64">CARD_3:64</a></i>;<br/>
<i><font color="Green" title="E47">A15</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">a</font> = 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">B</font>
<b>by </b><i><a class="txt" href="#E18:54_1"><i><font color="Green" title="E45">A13</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>
<br/>.= 
<font color="Maroon" title="c1">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">a</font> = <font color="Maroon" title="c11">a</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="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<i><font color="Green" title="E48">A16</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">b</font> = 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">B</font>
<b>by </b><i><a class="txt" href="#E20:54_1"><i><font color="Green" title="E46">A14</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>
<br/>.= 
<font color="Maroon" title="c1">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">b</font> = <font color="Maroon" title="c12">b</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E27:54_1"/><i><font color="Green" title="E49">A17</font></i>: 
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> &amp; <font color="Maroon" title="c13">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> )
 <b>by </b><i><a class="txt" href="#E17:54_1"><i><font color="Green" title="E44">A12</font></i></a>, <a class="txt" href="#E22:54_1"><i><font color="Green" title="E47">A15</font></i></a>, <a class="txt" href="#E24:54_1"><i><font color="Green" title="E48">A16</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">x</font> = <font color="Maroon" title="c15">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="txt" href="#E27:54_1"><i><font color="Green" title="E49">A17</font></i></a></i>;<br/>
<a NAME="E29:54_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E30:54_1"/><b>then </b><i><font color="Green" title="E50">A18</font></i>: 
<font color="Maroon" title="c13">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font>
 
<b>by </b><i><a class="txt" href="#E18:54_1"><i><font color="Green" title="E45">A13</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E31:54_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E32:54_1"/><b>then </b><i><font color="Green" title="E51">A19</font></i>: 
<font color="Maroon" title="c14">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font>
 
<b>by </b><i><a class="txt" href="#E20:54_1"><i><font color="Green" title="E46">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="E33:54_1"/><b>then </b>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E27:54_1"><i><font color="Green" title="E49">A17</font></i></a>, <a class="txt" href="#E30:54_1"><i><font color="Green" title="E50">A18</font></i></a>, <a class="ref" href="pencil_1.html#T2" target="_self" title="PENCIL_1:th.2">Th2</a></i>;<br/>
<a NAME="E34:54_1"/><b>then </b>
not <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>
 
<b>by </b><i><a class="ref" href="pencil_1.html#T4" target="_self" title="PENCIL_1:th.4">Th4</a></i>;<br/>
<a NAME="E35:54_1"/><b>then </b><i><font color="Green" title="E52">A20</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i</font>
 
<b>by </b><i><a class="txt" href="#E7:54_1"><i><font color="Green" title="E40">A4</font></i></a></i>;<br/>
<a NAME="E36:54_1"/><i><font color="Green" title="E53">A21</font></i>: 
<font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c7">C</font>
 
<b>by </b><i><a class="txt" href="#E11:54_1"><i><font color="Green" title="E41">A7</font></i></a>, <a class="txt" href="#E17:54_1"><i><font color="Green" title="E44">A12</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E37:54_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">C</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="E38:54_1"/><b>then </b><i><font color="Green" title="E54">A22</font></i>: 
<font color="Maroon" title="c13">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font>
 
<b>by </b><i><a class="txt" href="#E36:54_1"><i><font color="Green" title="E53">A21</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E39:54_1"/><i><font color="Green" title="E55">A23</font></i>: 
<font color="Maroon" title="c14">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c7">C</font>
 
<b>by </b><i><a class="txt" href="#E11:54_1"><i><font color="Green" title="E41">A7</font></i></a>, <a class="txt" href="#E17:54_1"><i><font color="Green" title="E44">A12</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E40:54_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">C</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="E41:54_1"/><b>then </b><i><font color="Green" title="E56">A24</font></i>: 
<font color="Maroon" title="c14">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font>
 
<b>by </b><i><a class="txt" href="#E39:54_1"><i><font color="Green" title="E55">A23</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E42:54_1"/><b>then </b>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E27:54_1"><i><font color="Green" title="E49">A17</font></i></a>, <a class="txt" href="#E38:54_1"><i><font color="Green" title="E54">A22</font></i></a>, <a class="ref" href="pencil_1.html#T2" target="_self" title="PENCIL_1:th.2">Th2</a></i>;<br/>
<a NAME="E43:54_1"/><b>then </b>
not <font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>
 
<b>by </b><i><a class="ref" href="pencil_1.html#T4" target="_self" title="PENCIL_1:th.4">Th4</a></i>;<br/>
<a NAME="E44:54_1"/><b>then </b><i><font color="Green" title="E57">A25</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font>
 
<b>by </b><i><a class="txt" href="#E14:54_1"><i><font color="Green" title="E43">A9</font></i></a></i>;<br/>
<i><font color="Green" title="E58">A26</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">B</font> = 
<font color="Maroon" title="c1">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">C</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/>
<a NAME="E46:54_1"/>
for <font color="Olive" title="b1">s</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">B</font> holds <br/><font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c17">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">B</font> implies <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font> )</span><br/>

<b>assume </b><a NAME="E1:54_1_6"/><i><font color="Green" title="E59">A27</font></i>: 
<font color="Maroon" title="c17">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c18">t</font> = <font color="Maroon" title="c17">s</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_1_6_1"/>
( <font color="Maroon" title="c18">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font> or <font color="Maroon" title="c17">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_6_1_1"><b>suppose </b></a><a NAME="E1:54_1_6_1_1"/><i><font color="Green" title="E60">A28</font></i>: 
<font color="Maroon" title="c18">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c19">Bt</font> = <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font> as   <a href="pencil_1.html#NM1" title="PENCIL_1:NM.1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K1" title="PENCIL_1:func.1">.</a> <font color="Maroon" title="c18">t</font></span>)</span> <b>by </b><i><a class="txt" href="#E6:54_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="#E35:54_1"><i><font color="Green" title="E52">A20</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">Ct</font> = <font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font> as   <a href="pencil_1.html#NM1" title="PENCIL_1:NM.1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K1" title="PENCIL_1:func.1">.</a> <font color="Maroon" title="c18">t</font></span>)</span> <b>by </b><i><a class="txt" href="#E13:54_1"><i><font color="Green" title="E42">A8</font></i></a>, <a class="txt" href="#E44:54_1"><i><font color="Green" title="E57">A25</font></i></a>, <a class="txt" href="#E1:54_1_6_1_1"><i><font color="Green" title="E60">A28</font></i></a></i>;<br/><a NAME="E4:54_1_6_1_1"/><i><font color="Green" title="E61">A29</font></i>: 
<font color="Maroon" title="c13">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">Bt</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c20">Ct</font>
 <b>by </b><i><a class="txt" href="#E30:54_1"><i><font color="Green" title="E50">A18</font></i></a>, <a class="txt" href="#E38:54_1"><i><font color="Green" title="E54">A22</font></i></a>, <a class="txt" href="#E1:54_1_6_1_1"><i><font color="Green" title="E60">A28</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E5:54_1_6_1_1"/>
<font color="Maroon" title="c14">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">Bt</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c20">Ct</font>
 <b>by </b><i><a class="txt" href="#E32:54_1"><i><font color="Green" title="E51">A19</font></i></a>, <a class="txt" href="#E41:54_1"><i><font color="Green" title="E56">A24</font></i></a>, <a class="txt" href="#E1:54_1_6_1_1"><i><font color="Green" title="E60">A28</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:54_1_6_1_1"/><b>then </b><i><font color="Green" title="E62">A30</font></i>: 
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">Bt</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c20">Ct</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E27:54_1"><i><font color="Green" title="E49">A17</font></i></a>, <a class="txt" href="#E1:54_1_6_1_1"><i><font color="Green" title="E60">A28</font></i></a>, <a class="txt" href="#E4:54_1_6_1_1"><i><font color="Green" title="E61">A29</font></i></a>, <a class="ref" href="pencil_1.html#T2" target="_self" title="PENCIL_1:th.2">Th2</a></i>;<br/><a NAME="E7:54_1_6_1_1"/>
<font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K1" title="PENCIL_1:func.1">.</a> <font color="Maroon" title="c18">t</font> is <a href="pencil_1.html#V6" title="PENCIL_1:attr.6">identifying_close_blocks</a>
 <b>by </b><i><a class="txt" href="#E1:54"><i><font color="Green" title="E37">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E8:54_1_6_1_1"/>
<font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font>
 <b>by </b><i><a class="txt" href="#E6:54_1_6_1_1"><i><font color="Green" title="E62">A30</font></i></a>, <a class="ref" href="pencil_1.html#D7" target="_self" title="PENCIL_1:def.7">Def7</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="54_1_6_1_2"><b>suppose </b></a><a NAME="E1:54_1_6_1_2"/>
<font color="Maroon" title="c17">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font></span><br/><div class="add"><a NAME="E2:54_1_6_1_2"/><b>then </b>
( not <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> )
 <b>by </b><i><a class="txt" href="#E7:54_1"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="#E35:54_1"><i><font color="Green" title="E52">A20</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c19">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:54_1_6_1_2"/><i><font color="Green" title="E60">A31</font></i>: 
<font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</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="c19">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="realset1.html#D4" title="REALSET1:def.4">REALSET1:def 4</a></i>;<br/><a NAME="E5:54_1_6_1_2"/>
<font color="Maroon" title="c13">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font>
 <b>by </b><i><a class="txt" href="#E18:54_1"><i><font color="Green" title="E45">A13</font></i></a>, <a class="txt" href="#E1:54_1_6"><i><font color="Green" title="E59">A27</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><a NAME="E6:54_1_6_1_2"/><b>then </b><i><font color="Green" title="E61">A32</font></i>: 
<font color="Maroon" title="c13">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y</font>
 <b>by </b><i><a class="txt" href="#E4:54_1_6_1_2"><i><font color="Green" title="E60">A31</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E7:54_1_6_1_2"/>
<font color="Maroon" title="c13">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font>
 <b>by </b><i><a class="txt" href="#E36:54_1"><i><font color="Green" title="E53">A21</font></i></a>, <a class="txt" href="#E45:54_1"><i><font color="Green" title="E58">A26</font></i></a>, <a class="txt" href="#E1:54_1_6"><i><font color="Green" title="E59">A27</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><b>hence </b><a NAME="E8:54_1_6_1_2"/>
<font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font>
 <b>by </b><i><a class="txt" href="#E4:54_1_6_1_2"><i><font color="Green" title="E60">A31</font></i></a>, <a class="txt" href="#E6:54_1_6_1_2"><i><font color="Green" title="E61">A32</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</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="E47:54_1"/>
<font color="Maroon" title="c3">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">l</font>
 <b>by </b><i><a class="txt" href="#E4:54_1"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="#E11:54_1"><i><font color="Green" title="E41">A7</font></i></a>, <a class="txt" href="#E45:54_1"><i><font color="Green" title="E58">A26</font></i></a>, <a class="ref" href="card_3.html#T38" title="CARD_3:th.38">CARD_3:38</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">k</font></span><br/>

<a NAME="E48:54_1"/>
for <font color="Olive" title="b1">s</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">C</font> holds <br/><font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c17">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">C</font> implies <font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font> )</span><br/>

<b>assume </b><a NAME="E1:54_1_7"/><i><font color="Green" title="E59">A33</font></i>: 
<font color="Maroon" title="c17">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c18">t</font> = <font color="Maroon" title="c17">s</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_1_7_1"/>
( <font color="Maroon" title="c18">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font> or <font color="Maroon" title="c17">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_7_1_1"><b>suppose </b></a><a NAME="E1:54_1_7_1_1"/><i><font color="Green" title="E60">A34</font></i>: 
<font color="Maroon" title="c18">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c19">Bt</font> = <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font> as   <a href="pencil_1.html#NM1" title="PENCIL_1:NM.1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K1" title="PENCIL_1:func.1">.</a> <font color="Maroon" title="c18">t</font></span>)</span> <b>by </b><i><a class="txt" href="#E6:54_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="#E35:54_1"><i><font color="Green" title="E52">A20</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">Ct</font> = <font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font> as   <a href="pencil_1.html#NM1" title="PENCIL_1:NM.1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K1" title="PENCIL_1:func.1">.</a> <font color="Maroon" title="c18">t</font></span>)</span> <b>by </b><i><a class="txt" href="#E13:54_1"><i><font color="Green" title="E42">A8</font></i></a>, <a class="txt" href="#E44:54_1"><i><font color="Green" title="E57">A25</font></i></a>, <a class="txt" href="#E1:54_1_7_1_1"><i><font color="Green" title="E60">A34</font></i></a></i>;<br/><a NAME="E4:54_1_7_1_1"/><i><font color="Green" title="E61">A35</font></i>: 
<font color="Maroon" title="c13">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">Bt</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c20">Ct</font>
 <b>by </b><i><a class="txt" href="#E30:54_1"><i><font color="Green" title="E50">A18</font></i></a>, <a class="txt" href="#E38:54_1"><i><font color="Green" title="E54">A22</font></i></a>, <a class="txt" href="#E1:54_1_7_1_1"><i><font color="Green" title="E60">A34</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E5:54_1_7_1_1"/>
<font color="Maroon" title="c14">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">Bt</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c20">Ct</font>
 <b>by </b><i><a class="txt" href="#E32:54_1"><i><font color="Green" title="E51">A19</font></i></a>, <a class="txt" href="#E41:54_1"><i><font color="Green" title="E56">A24</font></i></a>, <a class="txt" href="#E1:54_1_7_1_1"><i><font color="Green" title="E60">A34</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:54_1_7_1_1"/><b>then </b><i><font color="Green" title="E62">A36</font></i>: 
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">Bt</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c20">Ct</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E27:54_1"><i><font color="Green" title="E49">A17</font></i></a>, <a class="txt" href="#E1:54_1_7_1_1"><i><font color="Green" title="E60">A34</font></i></a>, <a class="txt" href="#E4:54_1_7_1_1"><i><font color="Green" title="E61">A35</font></i></a>, <a class="ref" href="pencil_1.html#T2" target="_self" title="PENCIL_1:th.2">Th2</a></i>;<br/><a NAME="E7:54_1_7_1_1"/>
<font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K1" title="PENCIL_1:func.1">.</a> <font color="Maroon" title="c18">t</font> is <a href="pencil_1.html#V6" title="PENCIL_1:attr.6">identifying_close_blocks</a>
 <b>by </b><i><a class="txt" href="#E1:54"><i><font color="Green" title="E37">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E8:54_1_7_1_1"/>
<font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font>
 <b>by </b><i><a class="txt" href="#E6:54_1_7_1_1"><i><font color="Green" title="E62">A36</font></i></a>, <a class="ref" href="pencil_1.html#D7" target="_self" title="PENCIL_1:def.7">Def7</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="54_1_7_1_2"><b>suppose </b></a><a NAME="E1:54_1_7_1_2"/>
<font color="Maroon" title="c17">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font></span><br/><div class="add"><a NAME="E2:54_1_7_1_2"/><b>then </b>
( not <font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> )
 <b>by </b><i><a class="txt" href="#E14:54_1"><i><font color="Green" title="E43">A9</font></i></a>, <a class="txt" href="#E44:54_1"><i><font color="Green" title="E57">A25</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c19">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:54_1_7_1_2"/><i><font color="Green" title="E60">A37</font></i>: 
<font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</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="c19">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="realset1.html#D4" title="REALSET1:def.4">REALSET1:def 4</a></i>;<br/><a NAME="E5:54_1_7_1_2"/>
<font color="Maroon" title="c13">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font>
 <b>by </b><i><a class="txt" href="#E36:54_1"><i><font color="Green" title="E53">A21</font></i></a>, <a class="txt" href="#E1:54_1_7"><i><font color="Green" title="E59">A33</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><a NAME="E6:54_1_7_1_2"/><b>then </b><i><font color="Green" title="E61">A38</font></i>: 
<font color="Maroon" title="c13">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y</font>
 <b>by </b><i><a class="txt" href="#E4:54_1_7_1_2"><i><font color="Green" title="E60">A37</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E7:54_1_7_1_2"/>
<font color="Maroon" title="c13">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font>
 <b>by </b><i><a class="txt" href="#E18:54_1"><i><font color="Green" title="E45">A13</font></i></a>, <a class="txt" href="#E45:54_1"><i><font color="Green" title="E58">A26</font></i></a>, <a class="txt" href="#E1:54_1_7"><i><font color="Green" title="E59">A33</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><b>hence </b><a NAME="E8:54_1_7_1_2"/>
<font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font>
 <b>by </b><i><a class="txt" href="#E4:54_1_7_1_2"><i><font color="Green" title="E60">A37</font></i></a>, <a class="txt" href="#E6:54_1_7_1_2"><i><font color="Green" title="E61">A38</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</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="E49:54_1"/>
<font color="Maroon" title="c4">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="#E4:54_1"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="#E11:54_1"><i><font color="Green" title="E41">A7</font></i></a>, <a class="txt" href="#E45:54_1"><i><font color="Green" title="E58">A26</font></i></a>, <a class="ref" href="card_3.html#T38" title="CARD_3:th.38">CARD_3:38</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:54"/>
 <a href="pencil_1.html#K5" title="PENCIL_1:func.5">Segre_Product</a> <font color="Maroon" title="c2">A</font> is <a href="pencil_1.html#V6" title="PENCIL_1:attr.6">identifying_close_blocks</a>
 <b>by </b><i><a class="ref" href="pencil_1.html#D7" target="_self" title="PENCIL_1:def.7">Def7</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>
