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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>for <font color="Olive" title="b1">e1</font>, <font color="Olive" title="b2">e2</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e1</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 <font color="Maroon" title="c2">Y</font> &amp; <font color="Olive" title="b2">e2</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 <font color="Maroon" title="c2">Y</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">e1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Olive" title="b2">e2</font> holds <br/> ex <font color="Olive" title="b5">F</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">X</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 &amp; <font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font> <a href="lattice5.html#R1" title="LATTICE5:pred.1">are_joint_by</a> <font color="Olive" title="b5">F</font>,<font color="Olive" title="b1">e1</font>,<font color="Olive" title="b2">e2</font> );<br/>
<b>set </b><font color="Maroon" title="c5">A</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">n</font> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] </span>}</span> ;<br/>
<a NAME="E1:24_1_1"/><i><font color="Green" title="E18">A4</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T16" title="EQREL_1:th.16">EQREL_1:16</a></i>;<br/>
<a NAME="E2:24_1_1"/><b>then </b>
 <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">X</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c3">e</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#T17" title="RELAT_2:th.17">RELAT_2:17</a></i>;<br/>
<a NAME="E3:24_1_1"/><b>then </b>
not <font color="Maroon" title="c3">e</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="#E3:24"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:24_1_1"/><i><font color="Green" title="E19">A5</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">e</font>
 <b>and </b><br/><a NAME="E6:24_1_1"/><i><font color="Green" title="E20">A6</font></i>: 
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D3" title="RELAT_1:def.3">RELAT_1:def 3</a></i>;<br/>
<a NAME="E7:24_1_1"/><i><font color="Green" title="E21">A7</font></i>: 
(  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">y</font> )
 
<b>by </b><i><a class="txt" href="#E6:24_1_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="relat_1.html#D10" title="RELAT_1:def.10">RELAT_1:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">e1</font>, <font color="Maroon" title="c9">e2</font> being   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E9:24_1_1"/><i><font color="Green" title="E22">A8</font></i>: 
( <font color="Maroon" title="c8">e1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">e</font> &amp; <font color="Maroon" title="c9">e2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">e</font> )
 ;<br/>
<a NAME="E10:24_1_1"/><i><font color="Green" title="E23">A9</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">e1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Maroon" title="c9">e2</font>
 
<b>by </b><i><a class="txt" href="#E5:24_1_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="#E9:24_1_1"><i><font color="Green" title="E22">A8</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">F</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E12:24_1_1"/><i><font color="Green" title="E24">A10</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o</font>
 <b>and </b><br/><a NAME="E13:24_1_1"/><i><font color="Green" title="E25">A11</font></i>: 
<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font> <a href="lattice5.html#R1" title="LATTICE5:pred.1">are_joint_by</a> <font color="Maroon" title="c10">F</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c9">e2</font>
 <b>by </b><i><a class="txt" href="#E2:24"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="#E5:24"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="#E9:24_1_1"><i><font color="Green" title="E22">A8</font></i></a></i>;<br/>
<a NAME="E14:24_1_1"/><i><font color="Green" title="E26">A12</font></i>: 
( <font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</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  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">F</font> holds <br/>( ( not <font color="Olive" title="b1">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">e1</font> ) &amp; ( <font color="Olive" title="b1">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">e2</font> ) ) ) )
 
<b>by </b><i><a class="txt" href="#E13:24_1_1"><i><font color="Green" title="E25">A11</font></i></a>, <a class="ref" href="lattice5.html#D3" target="_self" title="LATTICE5:def.3">Def3</a></i>;<br/>
<a NAME="E15:24_1_1"/>
<font color="Maroon" title="c4">o</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1_1_1"/><i><font color="Green" title="E27">A13</font></i>: 
not <font color="Maroon" title="c4">o</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:24_1_1_1"/><i><font color="Green" title="E28">A14</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">F</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/>
<a NAME="E3:24_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">F</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="#E12:24_1_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="#E1:24_1_1_1"><i><font color="Green" title="E27">A13</font></i></a></i>;<br/>
<a NAME="E4:24_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">F</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E5:24_1_1_1"/><b>then </b>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">F</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="#E2:24_1_1_1"><i><font color="Green" title="E28">A14</font></i></a>, <a class="ref" href="nat_1.html#T9" title="NAT_1:th.9">NAT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E6:24_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:24_1_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="#E5:24_1_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="#E7:24_1_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="#E14:24_1_1"><i><font color="Green" title="E26">A12</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c11">o'</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E17:24_1_1"/><i><font color="Green" title="E27">A15</font></i>: 
<font color="Maroon" title="c4">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">o'</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/>
<a NAME="E18:24_1_1"/>
<font color="Maroon" title="c11">o'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E20:24_1_1"/><i><font color="Green" title="E28">A16</font></i>: 
( <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">o'</font> &amp; ( for <font color="Olive" title="b1">e1</font>, <font color="Olive" title="b2">e2</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e1</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 <font color="Maroon" title="c2">Y</font> &amp; <font color="Olive" title="b2">e2</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 <font color="Maroon" title="c2">Y</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">e1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Olive" title="b2">e2</font> holds <br/> ex <font color="Olive" title="b5">F</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">X</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 &amp; <font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font> <a href="lattice5.html#R1" title="LATTICE5:pred.1">are_joint_by</a> <font color="Olive" title="b5">F</font>,<font color="Olive" title="b1">e1</font>,<font color="Olive" title="b2">e2</font> ) ) )
 <b>by </b><i><a class="txt" href="#E5:24"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="#E17:24_1_1"><i><font color="Green" title="E27">A15</font></i></a></i>;<br/>
<a NAME="E21:24_1_1"/><i><font color="Green" title="E29">A17</font></i>: 
<font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">n</font> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="#E20:24_1_1"><i><font color="Green" title="E28">A16</font></i></a></i>;<br/>
<a NAME="E22:24_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">n</font> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] </span>}</span>  is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>from </b><i><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></i>
<b>then reconsider </b><font color="Maroon" title="c13">A</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">n</font> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E21:24_1_1"><i><font color="Green" title="E29">A17</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c14">m</font> =  <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c13">A</font>;<br/>
<a NAME="E24:24_1_1"/>
 <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c13">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A</font>
 
<b>by </b><i><a class="ref" href="xxreal_2.html#D7" title="XXREAL_2:def.7">XXREAL_2:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">m'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E26:24_1_1"/><i><font color="Green" title="E30">A18</font></i>: 
( <font color="Maroon" title="c15">m'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c13">A</font> &amp; ( for <font color="Olive" title="b1">e1</font>, <font color="Olive" title="b2">e2</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e1</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 <font color="Maroon" title="c2">Y</font> &amp; <font color="Olive" title="b2">e2</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 <font color="Maroon" title="c2">Y</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">e1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Olive" title="b2">e2</font> holds <br/> ex <font color="Olive" title="b5">F</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">X</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">m'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 &amp; <font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font> <a href="lattice5.html#R1" title="LATTICE5:pred.1">are_joint_by</a> <font color="Olive" title="b5">F</font>,<font color="Olive" title="b1">e1</font>,<font color="Olive" title="b2">e2</font> ) ) )
 ;<br/>
<a NAME="E27:24_1_1"/>
 ex <font color="Olive" title="b1">e1</font>, <font color="Olive" title="b2">e2</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font> ex <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">e1</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 <font color="Maroon" title="c2">Y</font> &amp; <font color="Olive" title="b2">e2</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 <font color="Maroon" title="c2">Y</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">e1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Olive" title="b2">e2</font> &amp; ( for <font color="Olive" title="b5">F</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">X</font>  holds <br/>(  not  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c13">A</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or  not <font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font> <a href="lattice5.html#R1" title="LATTICE5:pred.1">are_joint_by</a> <font color="Olive" title="b5">F</font>,<font color="Olive" title="b1">e1</font>,<font color="Olive" title="b2">e2</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1_1_2"/><i><font color="Green" title="E31">A19</font></i>: 
for <font color="Olive" title="b1">e1</font>, <font color="Olive" title="b2">e2</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">e1</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 <font color="Maroon" title="c2">Y</font> or  not <font color="Olive" title="b2">e2</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 <font color="Maroon" title="c2">Y</font> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">e1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Olive" title="b2">e2</font> or  ex <font color="Olive" title="b5">F</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">X</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c13">A</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font> <a href="lattice5.html#R1" title="LATTICE5:pred.1">are_joint_by</a> <font color="Olive" title="b5">F</font>,<font color="Olive" title="b1">e1</font>,<font color="Olive" title="b2">e2</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c16">F</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:24_1_1_2"/><i><font color="Green" title="E32">A20</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c13">A</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font> <a href="lattice5.html#R1" title="LATTICE5:pred.1">are_joint_by</a> <font color="Maroon" title="c16">F</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c9">e2</font> )
 <b>by </b><i><a class="txt" href="#E2:24"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="#E9:24_1_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="#E10:24_1_1"><i><font color="Green" title="E23">A9</font></i></a></i>;<br/>
<a NAME="E4:24_1_1_2"/><i><font color="Green" title="E33">A21</font></i>: 
( <font color="Maroon" title="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> )
 
<b>by </b><i><a class="txt" href="#E3:24_1_1_2"><i><font color="Green" title="E32">A20</font></i></a>, <a class="ref" href="lattice5.html#D3" target="_self" title="LATTICE5:def.3">Def3</a></i>;<br/>
<a NAME="E5:24_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">F</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1_1_2_1"/><i><font color="Green" title="E34">A22</font></i>: 
not  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">F</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:24_1_1_2_1"/><i><font color="Green" title="E35">A23</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">F</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/>
<a NAME="E3:24_1_1_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">F</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="#E1:24_1_1_2_1"><i><font color="Green" title="E34">A22</font></i></a></i>;<br/>
<a NAME="E4:24_1_1_2_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">F</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E5:24_1_1_2_1"/><b>then </b>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">F</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="#E2:24_1_1_2_1"><i><font color="Green" title="E35">A23</font></i></a>, <a class="ref" href="nat_1.html#T9" title="NAT_1:th.9">NAT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E6:24_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:24_1_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="#E5:24_1_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="#E7:24_1_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="#E4:24_1_1_2"><i><font color="Green" title="E33">A21</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</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="E6:24_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c13">A</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="#E3:24_1_1_2"><i><font color="Green" title="E32">A20</font></i></a></i>;<br/>
<a NAME="E7:24_1_1_2"/><b>then </b><i><font color="Green" title="E34">A24</font></i>: 
 <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c13">A</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E8:24_1_1_2"/><b>then </b><i><font color="Green" title="E35">A25</font></i>: 
<span class="p1">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c13">A</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/>
<a NAME="E9:24_1_1_2"/>
 <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c13">A</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c13">A</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<a NAME="E10:24_1_1_2"/><b>then </b><i><font color="Green" title="E36">A26</font></i>: 
<span class="p1">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c13">A</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c13">A</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E11:24_1_1_2"/><i><font color="Green" title="E37">A27</font></i>: 
<span class="p1">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c13">A</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c13">A</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2
 
;<br/>
<a NAME="E12:24_1_1_2"/>
<span class="p1">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c13">A</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c13">A</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<b>by </b><i><a class="txt" href="#E7:24_1_1_2"><i><font color="Green" title="E34">A24</font></i></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></i>;<br/>
<a NAME="E13:24_1_1_2"/><b>then </b><i><font color="Green" title="E38">A28</font></i>: 
<span class="p1">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c13">A</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A</font>
 
<b>by </b><i><a class="txt" href="#E1:24_1_1_2"><i><font color="Green" title="E31">A19</font></i></a>, <a class="txt" href="#E11:24_1_1_2"><i><font color="Green" title="E37">A27</font></i></a></i>;<br/>
<a NAME="E14:24_1_1_2"/>
<span class="p1">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c13">A</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c13">A</font>
 
<b>by </b><i><a class="txt" href="#E8:24_1_1_2"><i><font color="Green" title="E35">A25</font></i></a>, <a class="txt" href="#E10:24_1_1_2"><i><font color="Green" title="E36">A26</font></i></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E15:24_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E13:24_1_1_2"><i><font color="Green" title="E38">A28</font></i></a>, <a class="ref" href="xxreal_2.html#D7" title="XXREAL_2:def.7">XXREAL_2: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>
<b>hence </b><a NAME="E28:24_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( ( for <font color="Olive" title="b2">e1</font>, <font color="Olive" title="b3">e2</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b4">x</font>, <font color="Olive" title="b5">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">e1</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 <font color="Maroon" title="c2">Y</font> &amp; <font color="Olive" title="b3">e2</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 <font color="Maroon" title="c2">Y</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">x</font>,<font color="Olive" title="b5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">e1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Olive" title="b3">e2</font> holds <br/> ex <font color="Olive" title="b6">F</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">X</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 &amp; <font color="Olive" title="b4">x</font>,<font color="Olive" title="b5">y</font> <a href="lattice5.html#R1" title="LATTICE5:pred.1">are_joint_by</a> <font color="Olive" title="b6">F</font>,<font color="Olive" title="b2">e1</font>,<font color="Olive" title="b3">e2</font> ) ) &amp;  ex <font color="Olive" title="b2">e1</font>, <font color="Olive" title="b3">e2</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font> ex <font color="Olive" title="b4">x</font>, <font color="Olive" title="b5">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">e1</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 <font color="Maroon" title="c2">Y</font> &amp; <font color="Olive" title="b3">e2</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 <font color="Maroon" title="c2">Y</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">x</font>,<font color="Olive" title="b5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">e1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Olive" title="b3">e2</font> &amp; ( for <font color="Olive" title="b6">F</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">X</font>  holds <br/>(  not  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or  not <font color="Olive" title="b4">x</font>,<font color="Olive" title="b5">y</font> <a href="lattice5.html#R1" title="LATTICE5:pred.1">are_joint_by</a> <font color="Olive" title="b6">F</font>,<font color="Olive" title="b2">e1</font>,<font color="Olive" title="b3">e2</font> ) ) ) )
 <b>by </b><i><a class="txt" href="#E26:24_1_1"><i><font color="Green" title="E30">A18</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>
