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<div class="add">

<b>let </b><font color="Maroon">A</font> be  non <a href="realset1.html#V1">trivial</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V4">full</a> <a href="lattice8.html#V1" target="_self">finitely_typed</a> <a href="lattice5.html#NM2">Sublattice</a> of  <a href="lattice5.html#K1">EqRelLATT</a> <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">e</font> be   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">A</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:15"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E2:15"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="partfun1.html#K6">id</a> <font color="Maroon">A</font>
 ;<br/>

<b>assume </b><a NAME="E3:15"/><i><font color="Green">E58</font></i>: 
 <a href="lattice5.html#K3">type_of</a> <font color="Maroon">L</font> <a href="xxreal_0.html#R1">&lt;=</a> 2
 ;<br/>

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

<b>assume </b><a NAME="E4:15"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c</font>
 ;<br/>

<a NAME="E5:15"/><i><font color="Green">E60</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K1">EqRelLATT</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D13">YELLOW_0:def 13</a></i>;<br/>
<a NAME="E6:15"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">A</font> <b>by </b><i>, <a class="ref" href="lattice5.html#T4">LATTICE5:4</a></i>;<br/>
<a NAME="E8:15"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">b</font> as   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">A</font> <b>by </b><i>, <a class="ref" href="lattice5.html#T4">LATTICE5:4</a></i>;<br/>
<a NAME="E10:15"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c'</font> = <font color="Maroon">c</font> as   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">A</font> <b>by </b><i>, <a class="ref" href="lattice5.html#T4">LATTICE5:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a''</font> = <font color="Maroon">a'</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K1">EqRelLATT</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">b''</font> = <font color="Maroon">b'</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K1">EqRelLATT</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">c''</font> = <font color="Maroon">c'</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K1">EqRelLATT</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i>, </i>;<br/>
<a NAME="E15:15"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">a''</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c''</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T60">YELLOW_0:60</a></i>;<br/>
<a NAME="E16:15"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">a'</font> <a href="lattice8.html#R1" target="_self">c=</a> <font color="Maroon">c'</font>
 
<b>by </b><i><a class="ref" href="lattice5.html#T6">LATTICE5:6</a></i>;<br/>
<a NAME="E17:15"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">b'</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">c'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b''</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c''</font> &amp; <font color="Maroon">b''</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c''</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c</font> )
 
<b>by </b><i><a class="ref" href="lattice5.html#T8">LATTICE5:8</a>, <a class="ref" href="yellow_0.html#T70">YELLOW_0:70</a></i>;<br/>
<a NAME="E18:15"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">a'</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a''</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b''</font> &amp; <font color="Maroon">a''</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b''</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="lattice5.html#T10">LATTICE5:10</a>, <a class="ref" href="yellow_0.html#T71">YELLOW_0:71</a></i>;<br/>
<a NAME="E19:15"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">a'</font> <a href="eqrel_1.html#K5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">c'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a''</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b''</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c''</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="lattice5.html#T10">LATTICE5:10</a></i>;<br/>
<a NAME="E20:15"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">a''</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b''</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c''</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a''</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b''</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c''</font>
 
<b>by </b><i>, <a class="ref" href="yellow11.html#T8">YELLOW11:8</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">a''</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b''</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c''</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a''</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b''</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c'</font>
<b>by </b><i><a class="ref" href="lattice5.html#T8">LATTICE5:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">b'</font></span>)</span> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">c'</font>
<b>by </b><i><a class="ref" href="lattice5.html#T10">LATTICE5:10</a></i>
;<br/>
<a NAME="E22:15"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">a'</font> <a href="eqrel_1.html#K5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">c'</font></span>)</span> <a href="lattice8.html#R1" target="_self">c=</a> <span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">b'</font></span>)</span> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">c'</font>
 
<b>by </b><i>, , <a class="ref" href="lattice5.html#T6">LATTICE5:6</a></i>;<br/>
<b>consider </b><font color="Maroon">AA</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E24:15"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> holds <br/><font color="Olive">e</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">AA</font>
 <b>and </b><br/><a NAME="E25:15"/><i><font color="Green">E79</font></i>: 
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">e1</font>, <font color="Olive">e2</font> being  <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">AA</font><br/> for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> &amp; <font color="Olive">e2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">e1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Olive">e2</font> holds <br/> ex <font color="Olive">F</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">AA</font> st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">x</font>,<font color="Olive">y</font> <a href="lattice5.html#R1">are_joint_by</a> <font color="Olive">F</font>,<font color="Olive">e1</font>,<font color="Olive">e2</font> )
 <b>by </b><i/>;<br/>
<a NAME="E26:15"/>
<font color="Maroon">e</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">AA</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E27:15"/><b>then </b><i><font color="Green">E80</font></i>: 
(  <a href="relat_1.html#K3">field</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp;  <a href="relat_1.html#K3">field</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">AA</font> )
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T16">EQREL_1:16</a></i>;<br/>
<a NAME="E28:15"/><i><font color="Green">E81</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">b'</font></span>)</span> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">c'</font> <a href="lattice8.html#R1" target="_self">c=</a> <font color="Maroon">a'</font> <a href="eqrel_1.html#K5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">c'</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice8.html#D1" target="_self">LATTICE8:def 1</a><br/>

<b>assume </b><a NAME="E1:15_2"/><i><font color="Green">E82</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">b'</font></span>)</span> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">c'</font>
 ;<br/>

<a NAME="E2:15_2"/><b>then </b><i><font color="Green">E83</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">a'</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">b'</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c'</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_2_1"/>
(  <a href="lattice5.html#K3">type_of</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> 2 or  <a href="lattice5.html#K3">type_of</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> 1 or  <a href="lattice5.html#K3">type_of</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> 0 )
 </span><b>by </b><i>, <a class="ref" href="cqc_the1.html#T3">CQC_THE1:3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_1_1"><b>suppose </b></a><a NAME="E1:15_2_1_1"/>
 <a href="lattice5.html#K3">type_of</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">F</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:15_2_1_1"/><i><font color="Green">E84</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K1">+</a> 2 &amp; <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="lattice5.html#R1">are_joint_by</a> <font color="Maroon">F</font>,<font color="Maroon">a'</font>,<font color="Maroon">b'</font> )
 <b>by </b><i>, , , , , <a class="ref" href="lattice5.html#D4">LATTICE5:def 4</a></i>;<br/><b>set </b><font color="Maroon">z1</font> = <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 2;<br/><b>set </b><font color="Maroon">z2</font> = <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 3;<br/><a NAME="E4:15_2_1_1"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 4 <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4 holds <br/>( ( not <font color="Olive">i</font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">a'</font> ) &amp; ( <font color="Olive">i</font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">b'</font> ) ) ) )
 <b>by </b><i>, <a class="ref" href="lattice5.html#D3">LATTICE5:def 3</a></i>;<br/><b>consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:15_2_1_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E7:15_2_1_1"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1
 <b>by </b><i/>;<br/><a NAME="E8:15_2_1_1"/><b>then </b><i><font color="Green">E87</font></i>: 
( not 1 is <a href="abian.html#V1">even</a> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">a'</font> )
 <b>by </b><i>, <a class="ref" href="lattice5.html#D3">LATTICE5:def 3</a></i>;<br/><b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E10:15_2_1_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E11:15_2_1_1"/>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i/>;<br/><a NAME="E12:15_2_1_1"/><b>then </b><i><font color="Green">E89</font></i>: 
( 2 is <a href="abian.html#V1">even</a> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">b'</font> )
 <b>by </b><i>, <a class="ref" href="lattice5.html#D3">LATTICE5:def 3</a></i>;<br/><b>consider </b><font color="Maroon">l</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E14:15_2_1_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E15:15_2_1_1"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">l</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 3
 <b>by </b><i/>;<br/><a NAME="E16:15_2_1_1"/><b>then </b><i><font color="Green">E91</font></i>: 
( not 3 is <a href="abian.html#V1">even</a> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default">3 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">a'</font> )
 <b>by </b><i>, <a class="ref" href="lattice5.html#D3">LATTICE5:def 3</a></i>;<br/><a NAME="E17:15_2_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">a'</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">a'</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">b'</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c'</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c'</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c'</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c'</font> )
 <b>by </b><i>, , , , , <a class="ref" href="eqrel_1.html#T12">EQREL_1:12</a></i>;<br/><a NAME="E18:15_2_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c'</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T13">EQREL_1:13</a></i>;<br/><a NAME="E19:15_2_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c'</font>
 <b>by </b><i>, , , <a class="ref" href="eqrel_1.html#T13">EQREL_1:13</a></i>;<br/><a NAME="E20:15_2_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c'</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T12">EQREL_1:12</a></i>;<br/><a NAME="E21:15_2_1_1"/><b>then </b><i><font color="Green">E92</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">b'</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">c'</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">BC</font> = <font color="Maroon">b'</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">c'</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K1">EqRelLATT</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i/>;<br/><a NAME="E23:15_2_1_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">a''</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a''</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">BC</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/><a NAME="E24:15_2_1_1"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">a'</font> <a href="eqrel_1.html#K5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">c'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a''</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">BC</font>
 <b>by </b><i><a class="ref" href="lattice5.html#T10">LATTICE5:10</a></i>;<br/><a NAME="E25:15_2_1_1"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">a'</font> <a href="lattice8.html#R1" target="_self">c=</a> <font color="Maroon">a'</font> <a href="eqrel_1.html#K5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">c'</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="lattice5.html#T6">LATTICE5:6</a></i>;<br/><a NAME="E26:15_2_1_1"/>
<font color="Maroon">BC</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">BC</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a''</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/><a NAME="E27:15_2_1_1"/><b>then </b>
<font color="Maroon">b'</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">c'</font> <a href="lattice8.html#R1" target="_self">c=</a> <font color="Maroon">a'</font> <a href="eqrel_1.html#K5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">c'</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E2:15"><i><font color="Green">E42</font></i></a>, <a class="ref" href="lattice5.html#T6">LATTICE5:6</a></i>;<br/><a NAME="E28:15_2_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">a'</font> <a href="eqrel_1.html#K5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">c'</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="eqrel_1.html#T13">EQREL_1:13</a></i>;<br/><b>hence </b><a NAME="E29:15_2_1_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">a'</font> <a href="eqrel_1.html#K5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">c'</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="lattice8.html#E1:15"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="eqrel_1.html#T13">EQREL_1:13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_1_2"><b>suppose </b></a><a NAME="E1:15_2_1_2"/>
 <a href="lattice5.html#K3">type_of</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">F</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:15_2_1_2"/><i><font color="Green">E125</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 2 &amp; <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="lattice5.html#R1">are_joint_by</a> <font color="Maroon">F</font>,<font color="Maroon">a'</font>,<font color="Maroon">b'</font> )
 <b>by </b><i>, , , , , <a class="ref" href="lattice5.html#D4">LATTICE5:def 4</a></i>;<br/><b>set </b><font color="Maroon">z1</font> = <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 2;<br/><a NAME="E4:15_2_1_2"/><i><font color="Green">E126</font></i>: 
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 3 holds <br/>( ( not <font color="Olive">i</font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">a'</font> ) &amp; ( <font color="Olive">i</font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">b'</font> ) ) ) )
 <b>by </b><i>, <a class="ref" href="lattice5.html#D3">LATTICE5:def 3</a></i>;<br/><b>consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:15_2_1_2"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E7:15_2_1_2"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1
 <b>by </b><i/>;<br/><a NAME="E8:15_2_1_2"/><b>then </b><i><font color="Green">E128</font></i>: 
( not 1 is <a href="abian.html#V1">even</a> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">a'</font> )
 <b>by </b><i>, <a class="ref" href="lattice5.html#D3">LATTICE5:def 3</a></i>;<br/><b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E10:15_2_1_2"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E11:15_2_1_2"/><i><font color="Green">E130</font></i>: 
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i/>;<br/><a NAME="E12:15_2_1_2"/><b>then </b><i><font color="Green">E131</font></i>: 
( 2 is <a href="abian.html#V1">even</a> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">b'</font> )
 <b>by </b><i>, <a class="ref" href="lattice5.html#D3">LATTICE5:def 3</a></i>;<br/><a NAME="E13:15_2_1_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">a'</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c'</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c'</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">b'</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c'</font> )
 <b>by </b><i>, , , , , , <a class="ref" href="eqrel_1.html#T12">EQREL_1:12</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E14:15_2_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c'</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T13">EQREL_1:13</a></i>;<br/><a NAME="E15:15_2_1_2"/><b>then </b><i><font color="Green">E132</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">b'</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">c'</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">BC</font> = <font color="Maroon">b'</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">c'</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K1">EqRelLATT</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i/>;<br/><a NAME="E17:15_2_1_2"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">a''</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a''</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">BC</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/><a NAME="E18:15_2_1_2"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">a'</font> <a href="eqrel_1.html#K5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">c'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a''</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">BC</font>
 <b>by </b><i><a class="ref" href="lattice5.html#T10">LATTICE5:10</a></i>;<br/><a NAME="E19:15_2_1_2"/><b>then </b><i><font color="Green">E135</font></i>: 
<font color="Maroon">a'</font> <a href="lattice8.html#R1" target="_self">c=</a> <font color="Maroon">a'</font> <a href="eqrel_1.html#K5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">c'</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="lattice5.html#T6">LATTICE5:6</a></i>;<br/><a NAME="E20:15_2_1_2"/>
<font color="Maroon">BC</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">BC</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a''</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/><a NAME="E21:15_2_1_2"/><b>then </b>
<font color="Maroon">b'</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">c'</font> <a href="lattice8.html#R1" target="_self">c=</a> <font color="Maroon">a'</font> <a href="eqrel_1.html#K5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">c'</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="lattice5.html#T6">LATTICE5:6</a></i>;<br/><b>hence </b><a NAME="E22:15_2_1_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">a'</font> <a href="eqrel_1.html#K5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">c'</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="lattice8.html#E1:15"><i><font color="Green">E39</font></i></a>, <a class="txt" href="lattice8.html#E2:15"><i><font color="Green">E42</font></i></a>, <a class="ref" href="eqrel_1.html#T13">EQREL_1:13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_1_3"><b>suppose </b></a><a NAME="E1:15_2_1_3"/>
 <a href="lattice5.html#K3">type_of</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">F</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:15_2_1_3"/><i><font color="Green">E136</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 2 &amp; <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="lattice5.html#R1">are_joint_by</a> <font color="Maroon">F</font>,<font color="Maroon">a'</font>,<font color="Maroon">b'</font> )
 <b>by </b><i>, , , , , <a class="ref" href="lattice5.html#D4">LATTICE5:def 4</a></i>;<br/><a NAME="E4:15_2_1_3"/><i><font color="Green">E137</font></i>: 
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 holds <br/>( ( not <font color="Olive">i</font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">a'</font> ) &amp; ( <font color="Olive">i</font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">b'</font> ) ) ) )
 <b>by </b><i>, <a class="ref" href="lattice5.html#D3">LATTICE5:def 3</a></i>;<br/><b>consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:15_2_1_3"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E7:15_2_1_3"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1
 <b>by </b><i/>;<br/><a NAME="E8:15_2_1_3"/><b>then </b><i><font color="Green">E139</font></i>: 
( not 1 is <a href="abian.html#V1">even</a> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">a'</font> )
 <b>by </b><i>, <a class="ref" href="lattice5.html#D3">LATTICE5:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">BC</font> = <font color="Maroon">b'</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">c'</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K1">EqRelLATT</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i/>;<br/><a NAME="E10:15_2_1_3"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">a''</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a''</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">BC</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/><a NAME="E11:15_2_1_3"/>
<font color="Maroon">a'</font> <a href="eqrel_1.html#K5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">c'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a''</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">BC</font>
 <b>by </b><i><a class="ref" href="lattice5.html#T10">LATTICE5:10</a></i>;<br/><a NAME="E12:15_2_1_3"/><b>then </b>
<font color="Maroon">a'</font> <a href="lattice8.html#R1" target="_self">c=</a> <font color="Maroon">a'</font> <a href="eqrel_1.html#K5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">c'</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="lattice5.html#T6">LATTICE5:6</a></i>;<br/><b>hence </b><a NAME="E13:15_2_1_3"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">a'</font> <a href="eqrel_1.html#K5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">c'</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<i><font color="Green">E141</font></i>: <font color="Maroon">a'</font> <a href="eqrel_1.html#K5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">c'</font></span>)</span> = 
<font color="Maroon">a''</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b''</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c''</font></span>)</span>
<b>by </b><i>, <a class="ref" href="lattice5.html#T10">LATTICE5:10</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T71">YELLOW_0:71</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a''</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b''</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c''</font>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T70">YELLOW_0:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a''</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b''</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c'</font>
<b>by </b><i><a class="ref" href="lattice5.html#T8">LATTICE5:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">b'</font></span>)</span> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">c'</font>
<b>by </b><i><a class="ref" href="lattice5.html#T10">LATTICE5:10</a></i>
;<br/>
<b>hence </b><a NAME="E31:15"/>
<font color="Maroon">a</font> <a href="lattice3.html#K10">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K10">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">c</font>
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
