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<b>let </b><font color="Maroon" title="c1">A</font> be  non <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a> <a href="lattice8.html#V1" title="LATTICE8:attr.1">finitely_typed</a> <a href="lattice5.html#NM2" title="LATTICE5:NM.2">Sublattice</a> of  <a href="lattice5.html#K1" title="LATTICE5:func.1">EqRelLATT</a> <font color="Maroon" title="c1">A</font>;<br/><b>let </b><font color="Maroon" title="c3">e</font> be   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">A</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:14"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c3">e</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">L</font>
 <b>and </b><br/><a NAME="E2:14"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c3">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">A</font>
 ;<br/>

<b>assume </b><a NAME="E3:14"/><i><font color="Green" title="E13">A3</font></i>: 
 <a href="lattice5.html#K3" title="LATTICE5:func.3">type_of</a> <font color="Maroon" title="c2">L</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2
 ;<br/>

<b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow11.html#D3" title="YELLOW11:def.3">YELLOW11:def 3</a><br/>

<b>assume </b><a NAME="E4:14"/><i><font color="Green" title="E14">A4</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">c</font>
 ;<br/>

<a NAME="E5:14"/><i><font color="Green" title="E15">A5</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K1" title="LATTICE5:func.1">EqRelLATT</a> <font color="Maroon" title="c1">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a></i>;<br/>
<a NAME="E6:14"/><i><font color="Green" title="E16">A6</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">a'</font> = <font color="Maroon" title="c4">a</font> as   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="lattice8.html#E5:14"><i><font color="Green" title="E15">A5</font></i></a>, <a class="ref" href="lattice5.html#T4" title="LATTICE5:th.4">LATTICE5:4</a></i>;<br/>
<a NAME="E8:14"/><i><font color="Green" title="E17">A7</font></i>: 
<font color="Maroon" title="c5">b</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">L</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">b'</font> = <font color="Maroon" title="c5">b</font> as   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="lattice8.html#E5:14"><i><font color="Green" title="E15">A5</font></i></a>, <a class="ref" href="lattice5.html#T4" title="LATTICE5:th.4">LATTICE5:4</a></i>;<br/>
<a NAME="E10:14"/><i><font color="Green" title="E18">A8</font></i>: 
<font color="Maroon" title="c6">c</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">L</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">c'</font> = <font color="Maroon" title="c6">c</font> as   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="lattice8.html#E5:14"><i><font color="Green" title="E15">A5</font></i></a>, <a class="ref" href="lattice5.html#T4" title="LATTICE5:th.4">LATTICE5:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">a''</font> = <font color="Maroon" title="c7">a'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K1" title="LATTICE5:func.1">EqRelLATT</a> <font color="Maroon" title="c1">A</font></span>)</span> <b>by </b><i><a class="txt" href="lattice8.html#E5:14"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="lattice8.html#E6:14"><i><font color="Green" title="E16">A6</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">b''</font> = <font color="Maroon" title="c8">b'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K1" title="LATTICE5:func.1">EqRelLATT</a> <font color="Maroon" title="c1">A</font></span>)</span> <b>by </b><i><a class="txt" href="lattice8.html#E5:14"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="lattice8.html#E8:14"><i><font color="Green" title="E17">A7</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">c''</font> = <font color="Maroon" title="c9">c'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K1" title="LATTICE5:func.1">EqRelLATT</a> <font color="Maroon" title="c1">A</font></span>)</span> <b>by </b><i><a class="txt" href="lattice8.html#E5:14"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="lattice8.html#E10:14"><i><font color="Green" title="E18">A8</font></i></a></i>;<br/>
<a NAME="E15:14"/><i><font color="Green" title="E19">A9</font></i>: 
<font color="Maroon" title="c10">a''</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">c''</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E4:14"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="yellow_0.html#T60" title="YELLOW_0:th.60">YELLOW_0:60</a></i>;<br/>
<a NAME="E16:14"/><b>then </b><i><font color="Green" title="E20">A10</font></i>: 
<font color="Maroon" title="c7">a'</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c9">c'</font>
 
<b>by </b><i><a class="ref" href="lattice5.html#T6" title="LATTICE5:th.6">LATTICE5:6</a></i>;<br/>
<a NAME="E17:14"/><i><font color="Green" title="E21">A11</font></i>: 
( <font color="Maroon" title="c8">b'</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c9">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b''</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">c''</font> &amp; <font color="Maroon" title="c11">b''</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">c''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c6">c</font> )
 
<b>by </b><i><a class="ref" href="lattice5.html#T8" title="LATTICE5:th.8">LATTICE5:8</a>, <a class="ref" href="yellow_0.html#T70" title="YELLOW_0:th.70">YELLOW_0:70</a></i>;<br/>
<a NAME="E18:14"/><i><font color="Green" title="E22">A12</font></i>: 
( <font color="Maroon" title="c7">a'</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Maroon" title="c8">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a''</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">b''</font> &amp; <font color="Maroon" title="c10">a''</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">b''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c5">b</font> )
 
<b>by </b><i><a class="ref" href="lattice5.html#T10" title="LATTICE5:th.10">LATTICE5:10</a>, <a class="ref" href="yellow_0.html#T71" title="YELLOW_0:th.71">YELLOW_0:71</a></i>;<br/>
<a NAME="E19:14"/><i><font color="Green" title="E23">A13</font></i>: 
<font color="Maroon" title="c7">a'</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c9">c'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a''</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b''</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">c''</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice8.html#E17:14"><i><font color="Green" title="E21">A11</font></i></a>, <a class="ref" href="lattice5.html#T10" title="LATTICE5:th.10">LATTICE5:10</a></i>;<br/>
<a NAME="E20:14"/><i><font color="Green" title="E24">A14</font></i>: 
<font color="Maroon" title="c10">a''</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b''</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">c''</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a''</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">b''</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">c''</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E15:14"><i><font color="Green" title="E19">A9</font></i></a>, <a class="ref" href="yellow11.html#T8" title="YELLOW11:th.8">YELLOW11:8</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">a''</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">b''</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">c''</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">a''</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">b''</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">c'</font>
<b>by </b><i><a class="ref" href="lattice5.html#T8" title="LATTICE5:th.8">LATTICE5:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a'</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Maroon" title="c8">b'</font></span>)</span> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c9">c'</font>
<b>by </b><i><a class="ref" href="lattice5.html#T10" title="LATTICE5:th.10">LATTICE5:10</a></i>
;<br/>
<a NAME="E22:14"/><b>then </b><i><font color="Green" title="E25">A15</font></i>: 
<font color="Maroon" title="c7">a'</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c9">c'</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a'</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Maroon" title="c8">b'</font></span>)</span> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c9">c'</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E19:14"><i><font color="Green" title="E23">A13</font></i></a>, <a class="txt" href="lattice8.html#E20:14"><i><font color="Green" title="E24">A14</font></i></a>, <a class="ref" href="lattice5.html#T6" title="LATTICE5:th.6">LATTICE5:6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">AA</font> being  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> <b> such that </b><br/><a NAME="E24:14"/><i><font color="Green" title="E26">A16</font></i>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</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">L</font> holds <br/><font color="Olive" title="b1">e</font> is   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c13">AA</font>
 <b>and </b><br/><a NAME="E25:14"/><i><font color="Green" title="E27">A17</font></i>: 
 ex <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 <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="c13">AA</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">L</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">L</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="c13">AA</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" title="b1">i</font> &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> )
 <b>by </b><i><a class="ref" href="lattice8.html#D2" target="_self" title="LATTICE8:def.2">Def2</a></i>;<br/>
<a NAME="E26:14"/>
<font color="Maroon" title="c3">e</font> is   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c13">AA</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E1:14"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="lattice8.html#E24:14"><i><font color="Green" title="E26">A16</font></i></a></i>;<br/>
<a NAME="E27:14"/><b>then </b><i><font color="Green" title="E28">A18</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">A</font> &amp;  <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="c13">AA</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="E28:14"/><i><font color="Green" title="E29">A19</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a'</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Maroon" title="c8">b'</font></span>)</span> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c9">c'</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c7">a'</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c9">c'</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">x</font>, <font color="Maroon" title="c15">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="relset_1.html#D2" title="RELSET_1:def.2">RELSET_1:def 2</a><br/>

<b>assume </b><a NAME="E1:14_2"/><i><font color="Green" title="E30">A20</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a'</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Maroon" title="c8">b'</font></span>)</span> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c9">c'</font>
 ;<br/>

<a NAME="E2:14_2"/><b>then </b><i><font color="Green" title="E31">A21</font></i>: 
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a'</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Maroon" title="c8">b'</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">c'</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_2_1"/>
(  <a href="lattice5.html#K3" title="LATTICE5:func.3">type_of</a> <font color="Maroon" title="c2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or  <a href="lattice5.html#K3" title="LATTICE5:func.3">type_of</a> <font color="Maroon" title="c2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="lattice5.html#K3" title="LATTICE5:func.3">type_of</a> <font color="Maroon" title="c2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 </span><b>by </b><i><a class="txt" href="lattice8.html#E3:14"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="nat_1.html#T27" title="NAT_1:th.27">NAT_1:27</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1_1"><b>suppose </b></a><a NAME="E1:14_2_1_1"/>
 <a href="lattice5.html#K3" title="LATTICE5:func.3">type_of</a> <font color="Maroon" title="c2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><div class="add"><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">A</font><b> such that </b><br/><a NAME="E3:14_2_1_1"/><i><font color="Green" title="E32">A22</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> 2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 &amp; <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">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="c7">a'</font>,<font color="Maroon" title="c8">b'</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E1:14"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="lattice8.html#E2:14"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="lattice8.html#E25:14"><i><font color="Green" title="E27">A17</font></i></a>, <a class="txt" href="lattice8.html#E27:14"><i><font color="Green" title="E28">A18</font></i></a>, <a class="txt" href="lattice8.html#E2:14_2"><i><font color="Green" title="E31">A21</font></i></a>, <a class="ref" href="lattice5.html#D4" title="LATTICE5:def.4">LATTICE5:def 4</a></i>;<br/><b>set </b><font color="Maroon" title="c17">z1</font> = <font color="Maroon" title="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2;<br/><b>set </b><font color="Maroon" title="c18">z2</font> = <font color="Maroon" title="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3;<br/><a NAME="E4:14_2_1_1"/><i><font color="Green" title="E33">A23</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="c14">x</font> &amp; <font color="Maroon" title="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">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> 4 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="c16">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="c16">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#K1" title="NAT_1:func.1">+</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="c7">a'</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="c16">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="c16">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#K1" title="NAT_1:func.1">+</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">b'</font> ) ) ) )
 <b>by </b><i><a class="txt" href="lattice8.html#E3:14_2_1_1"><i><font color="Green" title="E32">A22</font></i></a>, <a class="ref" href="lattice5.html#D3" title="LATTICE5:def.3">LATTICE5:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c19">j</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="E6:14_2_1_1"/><i><font color="Green" title="E34">A24</font></i>: 
<font color="Maroon" title="c19">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><a NAME="E7:14_2_1_1"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c19">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="lattice8.html#E6:14_2_1_1"><i><font color="Green" title="E34">A24</font></i></a></i>;<br/><a NAME="E8:14_2_1_1"/><b>then </b><i><font color="Green" title="E35">A25</font></i>: 
( not 1 is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <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="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c7">a'</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E3:14_2_1_1"><i><font color="Green" title="E32">A22</font></i></a>, <a class="ref" href="lattice5.html#D3" title="LATTICE5:def.3">LATTICE5:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c20">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="E10:14_2_1_1"/><i><font color="Green" title="E36">A26</font></i>: 
<font color="Maroon" title="c20">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E11:14_2_1_1"/>
2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c20">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <b>by </b><i><a class="txt" href="lattice8.html#E10:14_2_1_1"><i><font color="Green" title="E36">A26</font></i></a></i>;<br/><a NAME="E12:14_2_1_1"/><b>then </b><i><font color="Green" title="E37">A27</font></i>: 
( 2 is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <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="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">b'</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E3:14_2_1_1"><i><font color="Green" title="E32">A22</font></i></a>, <a class="ref" href="lattice5.html#D3" title="LATTICE5:def.3">LATTICE5:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c21">l</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="E14:14_2_1_1"/><i><font color="Green" title="E38">A28</font></i>: 
<font color="Maroon" title="c21">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E15:14_2_1_1"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c21">l</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 <b>by </b><i><a class="txt" href="lattice8.html#E14:14_2_1_1"><i><font color="Green" title="E38">A28</font></i></a></i>;<br/><a NAME="E16:14_2_1_1"/><b>then </b><i><font color="Green" title="E39">A29</font></i>: 
( not 3 is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <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="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default">3 <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c7">a'</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E3:14_2_1_1"><i><font color="Green" title="E32">A22</font></i></a>, <a class="ref" href="lattice5.html#D3" title="LATTICE5:def.3">LATTICE5:def 3</a></i>;<br/><a NAME="E17:14_2_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</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="c7">a'</font> &amp; <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="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3</span>)</span>,<font color="Maroon" title="c15">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="c7">a'</font> &amp; <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="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3</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">b'</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c15">y</font>,<font color="Maroon" title="c14">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">c'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</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">c'</font> &amp; <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="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3</span>)</span>,<font color="Maroon" title="c15">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="c9">c'</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">c'</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E16:14"><i><font color="Green" title="E20">A10</font></i></a>, <a class="txt" href="lattice8.html#E2:14_2"><i><font color="Green" title="E31">A21</font></i></a>, <a class="txt" href="lattice8.html#E4:14_2_1_1"><i><font color="Green" title="E33">A23</font></i></a>, <a class="txt" href="lattice8.html#E8:14_2_1_1"><i><font color="Green" title="E35">A25</font></i></a>, <a class="txt" href="lattice8.html#E12:14_2_1_1"><i><font color="Green" title="E37">A27</font></i></a>, <a class="ref" href="eqrel_1.html#T12" title="EQREL_1:th.12">EQREL_1:12</a></i>;<br/><a NAME="E18:14_2_1_1"/><b>then </b>
<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="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3</span>)</span>,<font color="Maroon" title="c14">x</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="c9">c'</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T13" title="EQREL_1:th.13">EQREL_1:13</a></i>;<br/><a NAME="E19:14_2_1_1"/><b>then </b>
<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="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</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">c'</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E16:14"><i><font color="Green" title="E20">A10</font></i></a>, <a class="txt" href="lattice8.html#E4:14_2_1_1"><i><font color="Green" title="E33">A23</font></i></a>, <a class="txt" href="lattice8.html#E8:14_2_1_1"><i><font color="Green" title="E35">A25</font></i></a>, <a class="ref" href="eqrel_1.html#T13" title="EQREL_1:th.13">EQREL_1:13</a></i>;<br/><a NAME="E20:14_2_1_1"/><b>then </b>
<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="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3</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">c'</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T12" title="EQREL_1:th.12">EQREL_1:12</a></i>;<br/><a NAME="E21:14_2_1_1"/><b>then </b><i><font color="Green" title="E40">A30</font></i>: 
<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="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3</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">b'</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c9">c'</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E12:14_2_1_1"><i><font color="Green" title="E37">A27</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c22">BC</font> = <font color="Maroon" title="c8">b'</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c9">c'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K1" title="LATTICE5:func.1">EqRelLATT</a> <font color="Maroon" title="c1">A</font></span>)</span> <b>by </b><i><a class="txt" href="lattice8.html#E17:14"><i><font color="Green" title="E21">A11</font></i></a></i>;<br/><a NAME="E23:14_2_1_1"/><i><font color="Green" title="E41">A31</font></i>: 
<font color="Maroon" title="c10">a''</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">a''</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c22">BC</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>;<br/><a NAME="E24:14_2_1_1"/><i><font color="Green" title="E42">A32</font></i>: 
<font color="Maroon" title="c7">a'</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c9">c'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a''</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c22">BC</font>
 <b>by </b><i><a class="ref" href="lattice5.html#T10" title="LATTICE5:th.10">LATTICE5:10</a></i>;<br/><a NAME="E25:14_2_1_1"/><b>then </b><i><font color="Green" title="E43">A33</font></i>: 
<font color="Maroon" title="c7">a'</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c7">a'</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c9">c'</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E23:14_2_1_1"><i><font color="Green" title="E41">A31</font></i></a>, <a class="ref" href="lattice5.html#T6" title="LATTICE5:th.6">LATTICE5:6</a></i>;<br/><a NAME="E26:14_2_1_1"/>
<font color="Maroon" title="c22">BC</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c22">BC</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">a''</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>;<br/><a NAME="E27:14_2_1_1"/><b>then </b>
<font color="Maroon" title="c8">b'</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c9">c'</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c7">a'</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c9">c'</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E24:14_2_1_1"><i><font color="Green" title="E42">A32</font></i></a>, <a class="ref" href="lattice5.html#T6" title="LATTICE5:th.6">LATTICE5:6</a></i>;<br/><a NAME="E28:14_2_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3</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="c7">a'</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c9">c'</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E4:14_2_1_1"><i><font color="Green" title="E33">A23</font></i></a>, <a class="txt" href="lattice8.html#E8:14_2_1_1"><i><font color="Green" title="E35">A25</font></i></a>, <a class="txt" href="lattice8.html#E21:14_2_1_1"><i><font color="Green" title="E40">A30</font></i></a>, <a class="txt" href="lattice8.html#E25:14_2_1_1"><i><font color="Green" title="E43">A33</font></i></a>, <a class="ref" href="eqrel_1.html#T13" title="EQREL_1:th.13">EQREL_1:13</a></i>;<br/><b>hence </b><a NAME="E29:14_2_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">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="c7">a'</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c9">c'</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E4:14_2_1_1"><i><font color="Green" title="E33">A23</font></i></a>, <a class="txt" href="lattice8.html#E16:14_2_1_1"><i><font color="Green" title="E39">A29</font></i></a>, <a class="txt" href="lattice8.html#E25:14_2_1_1"><i><font color="Green" title="E43">A33</font></i></a>, <a class="ref" href="eqrel_1.html#T13" title="EQREL_1:th.13">EQREL_1:13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1_2"><b>suppose </b></a><a NAME="E1:14_2_1_2"/>
 <a href="lattice5.html#K3" title="LATTICE5:func.3">type_of</a> <font color="Maroon" title="c2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><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">A</font><b> such that </b><br/><a NAME="E3:14_2_1_2"/><i><font color="Green" title="E32">A34</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> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 &amp; <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">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="c7">a'</font>,<font color="Maroon" title="c8">b'</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E1:14"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="lattice8.html#E2:14"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="lattice8.html#E25:14"><i><font color="Green" title="E27">A17</font></i></a>, <a class="txt" href="lattice8.html#E27:14"><i><font color="Green" title="E28">A18</font></i></a>, <a class="txt" href="lattice8.html#E2:14_2"><i><font color="Green" title="E31">A21</font></i></a>, <a class="ref" href="lattice5.html#D4" title="LATTICE5:def.4">LATTICE5:def 4</a></i>;<br/><b>set </b><font color="Maroon" title="c17">z1</font> = <font color="Maroon" title="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2;<br/><a NAME="E4:14_2_1_2"/><i><font color="Green" title="E33">A35</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="c14">x</font> &amp; <font color="Maroon" title="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">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> 3 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="c16">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="c16">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#K1" title="NAT_1:func.1">+</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="c7">a'</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="c16">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="c16">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#K1" title="NAT_1:func.1">+</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">b'</font> ) ) ) )
 <b>by </b><i><a class="txt" href="lattice8.html#E3:14_2_1_2"><i><font color="Green" title="E32">A34</font></i></a>, <a class="ref" href="lattice5.html#D3" title="LATTICE5:def.3">LATTICE5:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c18">j</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="E6:14_2_1_2"/><i><font color="Green" title="E34">A36</font></i>: 
<font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><a NAME="E7:14_2_1_2"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c18">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="lattice8.html#E6:14_2_1_2"><i><font color="Green" title="E34">A36</font></i></a></i>;<br/><a NAME="E8:14_2_1_2"/><b>then </b><i><font color="Green" title="E35">A37</font></i>: 
( not 1 is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <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="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c7">a'</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E3:14_2_1_2"><i><font color="Green" title="E32">A34</font></i></a>, <a class="ref" href="lattice5.html#D3" title="LATTICE5:def.3">LATTICE5:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c19">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="E10:14_2_1_2"/><i><font color="Green" title="E36">A38</font></i>: 
<font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E11:14_2_1_2"/><i><font color="Green" title="E37">A39</font></i>: 
2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <b>by </b><i><a class="txt" href="lattice8.html#E10:14_2_1_2"><i><font color="Green" title="E36">A38</font></i></a></i>;<br/><a NAME="E12:14_2_1_2"/><b>then </b><i><font color="Green" title="E38">A40</font></i>: 
( 2 is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <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="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">b'</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E3:14_2_1_2"><i><font color="Green" title="E32">A34</font></i></a>, <a class="ref" href="lattice5.html#D3" title="LATTICE5:def.3">LATTICE5:def 3</a></i>;<br/><a NAME="E13:14_2_1_2"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</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="c7">a'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</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">c'</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">c'</font> &amp; <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="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span>,<font color="Maroon" title="c15">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">b'</font> &amp; <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="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span>,<font color="Maroon" title="c14">x</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="c9">c'</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E16:14"><i><font color="Green" title="E20">A10</font></i></a>, <a class="txt" href="lattice8.html#E1:14_2"><i><font color="Green" title="E30">A20</font></i></a>, <a class="txt" href="lattice8.html#E3:14_2_1_2"><i><font color="Green" title="E32">A34</font></i></a>, <a class="txt" href="lattice8.html#E4:14_2_1_2"><i><font color="Green" title="E33">A35</font></i></a>, <a class="txt" href="lattice8.html#E8:14_2_1_2"><i><font color="Green" title="E35">A37</font></i></a>, <a class="txt" href="lattice8.html#E11:14_2_1_2"><i><font color="Green" title="E37">A39</font></i></a>, <a class="ref" href="eqrel_1.html#T12" title="EQREL_1:th.12">EQREL_1:12</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E14:14_2_1_2"/><b>then </b>
<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="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span>,<font color="Maroon" title="c15">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="c9">c'</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T13" title="EQREL_1:th.13">EQREL_1:13</a></i>;<br/><a NAME="E15:14_2_1_2"/><b>then </b><i><font color="Green" title="E39">A41</font></i>: 
<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="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span>,<font color="Maroon" title="c15">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">b'</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c9">c'</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E4:14_2_1_2"><i><font color="Green" title="E33">A35</font></i></a>, <a class="txt" href="lattice8.html#E12:14_2_1_2"><i><font color="Green" title="E38">A40</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">BC</font> = <font color="Maroon" title="c8">b'</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c9">c'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K1" title="LATTICE5:func.1">EqRelLATT</a> <font color="Maroon" title="c1">A</font></span>)</span> <b>by </b><i><a class="txt" href="lattice8.html#E17:14"><i><font color="Green" title="E21">A11</font></i></a></i>;<br/><a NAME="E17:14_2_1_2"/><i><font color="Green" title="E40">A42</font></i>: 
<font color="Maroon" title="c10">a''</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">a''</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c20">BC</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>;<br/><a NAME="E18:14_2_1_2"/><i><font color="Green" title="E41">A43</font></i>: 
<font color="Maroon" title="c7">a'</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c9">c'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a''</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c20">BC</font>
 <b>by </b><i><a class="ref" href="lattice5.html#T10" title="LATTICE5:th.10">LATTICE5:10</a></i>;<br/><a NAME="E19:14_2_1_2"/><b>then </b><i><font color="Green" title="E42">A44</font></i>: 
<font color="Maroon" title="c7">a'</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c7">a'</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c9">c'</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E17:14_2_1_2"><i><font color="Green" title="E40">A42</font></i></a>, <a class="ref" href="lattice5.html#T6" title="LATTICE5:th.6">LATTICE5:6</a></i>;<br/><a NAME="E20:14_2_1_2"/>
<font color="Maroon" title="c20">BC</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c20">BC</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">a''</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>;<br/><a NAME="E21:14_2_1_2"/><b>then </b>
<font color="Maroon" title="c8">b'</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c9">c'</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c7">a'</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c9">c'</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E18:14_2_1_2"><i><font color="Green" title="E41">A43</font></i></a>, <a class="ref" href="lattice5.html#T6" title="LATTICE5:th.6">LATTICE5:6</a></i>;<br/><b>hence </b><a NAME="E22:14_2_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">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="c7">a'</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c9">c'</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E4:14_2_1_2"><i><font color="Green" title="E33">A35</font></i></a>, <a class="txt" href="lattice8.html#E8:14_2_1_2"><i><font color="Green" title="E35">A37</font></i></a>, <a class="txt" href="lattice8.html#E15:14_2_1_2"><i><font color="Green" title="E39">A41</font></i></a>, <a class="txt" href="lattice8.html#E19:14_2_1_2"><i><font color="Green" title="E42">A44</font></i></a>, <a class="ref" href="eqrel_1.html#T13" title="EQREL_1:th.13">EQREL_1:13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1_3"><b>suppose </b></a><a NAME="E1:14_2_1_3"/>
 <a href="lattice5.html#K3" title="LATTICE5:func.3">type_of</a> <font color="Maroon" title="c2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><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">A</font><b> such that </b><br/><a NAME="E3:14_2_1_3"/><i><font color="Green" title="E32">A45</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> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 &amp; <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">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="c7">a'</font>,<font color="Maroon" title="c8">b'</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E1:14"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="lattice8.html#E2:14"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="lattice8.html#E25:14"><i><font color="Green" title="E27">A17</font></i></a>, <a class="txt" href="lattice8.html#E27:14"><i><font color="Green" title="E28">A18</font></i></a>, <a class="txt" href="lattice8.html#E2:14_2"><i><font color="Green" title="E31">A21</font></i></a>, <a class="ref" href="lattice5.html#D4" title="LATTICE5:def.4">LATTICE5:def 4</a></i>;<br/><a NAME="E4:14_2_1_3"/><i><font color="Green" title="E33">A46</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="c14">x</font> &amp; <font color="Maroon" title="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">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> 2 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="c16">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="c16">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#K1" title="NAT_1:func.1">+</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="c7">a'</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="c16">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="c16">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#K1" title="NAT_1:func.1">+</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">b'</font> ) ) ) )
 <b>by </b><i><a class="txt" href="lattice8.html#E3:14_2_1_3"><i><font color="Green" title="E32">A45</font></i></a>, <a class="ref" href="lattice5.html#D3" title="LATTICE5:def.3">LATTICE5:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c17">j</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="E6:14_2_1_3"/><i><font color="Green" title="E34">A47</font></i>: 
<font color="Maroon" title="c17">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><a NAME="E7:14_2_1_3"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c17">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="lattice8.html#E6:14_2_1_3"><i><font color="Green" title="E34">A47</font></i></a></i>;<br/><a NAME="E8:14_2_1_3"/><b>then </b><i><font color="Green" title="E35">A48</font></i>: 
( not 1 is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <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="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c7">a'</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E3:14_2_1_3"><i><font color="Green" title="E32">A45</font></i></a>, <a class="ref" href="lattice5.html#D3" title="LATTICE5:def.3">LATTICE5:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">BC</font> = <font color="Maroon" title="c8">b'</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c9">c'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K1" title="LATTICE5:func.1">EqRelLATT</a> <font color="Maroon" title="c1">A</font></span>)</span> <b>by </b><i><a class="txt" href="lattice8.html#E17:14"><i><font color="Green" title="E21">A11</font></i></a></i>;<br/><a NAME="E10:14_2_1_3"/><i><font color="Green" title="E36">A49</font></i>: 
<font color="Maroon" title="c10">a''</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">a''</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c18">BC</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>;<br/><a NAME="E11:14_2_1_3"/>
<font color="Maroon" title="c7">a'</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c9">c'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a''</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c18">BC</font>
 <b>by </b><i><a class="ref" href="lattice5.html#T10" title="LATTICE5:th.10">LATTICE5:10</a></i>;<br/><a NAME="E12:14_2_1_3"/><b>then </b>
<font color="Maroon" title="c7">a'</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c7">a'</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c9">c'</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E10:14_2_1_3"><i><font color="Green" title="E36">A49</font></i></a>, <a class="ref" href="lattice5.html#T6" title="LATTICE5:th.6">LATTICE5:6</a></i>;<br/><b>hence </b><a NAME="E13:14_2_1_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">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="c7">a'</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c9">c'</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E4:14_2_1_3"><i><font color="Green" title="E33">A46</font></i></a>, <a class="txt" href="lattice8.html#E8:14_2_1_3"><i><font color="Green" title="E35">A48</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<i><font color="Green" title="E30">A50</font></i>: <font color="Maroon" title="c7">a'</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c9">c'</font></span>)</span> = 
<font color="Maroon" title="c10">a''</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b''</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">c''</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice8.html#E17:14"><i><font color="Green" title="E21">A11</font></i></a>, <a class="ref" href="lattice5.html#T10" title="LATTICE5:th.10">LATTICE5:10</a></i>
<br/>.= 
<font color="Maroon" title="c4">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c6">c</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice8.html#E17:14"><i><font color="Green" title="E21">A11</font></i></a>, <a class="ref" href="yellow_0.html#T71" title="YELLOW_0:th.71">YELLOW_0:71</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c6">c</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">a''</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">b''</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">c''</font>
<b>by </b><i><a class="txt" href="lattice8.html#E18:14"><i><font color="Green" title="E22">A12</font></i></a>, <a class="ref" href="yellow_0.html#T70" title="YELLOW_0:th.70">YELLOW_0:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">a''</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">b''</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">c'</font>
<b>by </b><i><a class="ref" href="lattice5.html#T8" title="LATTICE5:th.8">LATTICE5:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a'</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Maroon" title="c8">b'</font></span>)</span> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c9">c'</font>
<b>by </b><i><a class="ref" href="lattice5.html#T10" title="LATTICE5:th.10">LATTICE5:10</a></i>
;<br/>
<b>hence </b><a NAME="E31:14"/>
<font color="Maroon" title="c4">a</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c6">c</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E22:14"><i><font color="Green" title="E25">A15</font></i></a>, <a class="txt" href="lattice8.html#E28:14"><i><font color="Green" title="E29">A19</font></i></a>, <a class="txt" href="lattice8.html#E29:14"><i><font color="Green" title="E30">A50</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div>
