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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="lattices.html#L3">LattStr</a> ;<br/>

<b>assume </b><a NAME="E1:4"/><i><font color="Green">E3</font></i>: 
 <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>1</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>2</sub></font> #)
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E2:4"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>thus </b><a NAME="E3:4"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">c<sub>5</sub></font> implies <font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">c<sub>5</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:4_1"/><i><font color="Green">E5</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R1">[=</a> <font color="Olive">b<sub>1</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a><br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="quantal1.html#E1:4"><i><font color="Green">E3</font></i></a></i>;<br/>
<b>assume </b><a NAME="E3:4_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E4:4_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R1">[=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="quantal1.html#E1:4_1"><i><font color="Green">E5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:4_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#R1">[=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="quantal1.html#E1:4"><i><font color="Green">E3</font></i></a>, <a class="txt" href="quantal1.html#E2:4"><i><font color="Green">E4</font></i></a>, <a class="ref" href="quantal1.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:4"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">c<sub>5</sub></font> implies <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">c<sub>5</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:4_2"/><i><font color="Green">E5</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#R1">[=</a> <font color="Olive">b<sub>1</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a><br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="quantal1.html#E1:4"><i><font color="Green">E3</font></i></a></i>;<br/>
<b>assume </b><a NAME="E3:4_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E4:4_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#R1">[=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="quantal1.html#E1:4_2"><i><font color="Green">E5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:4_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R1">[=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="quantal1.html#E1:4"><i><font color="Green">E3</font></i></a>, <a class="txt" href="quantal1.html#E2:4"><i><font color="Green">E4</font></i></a>, <a class="ref" href="quantal1.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E5:4"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#NR8" target="_self">is_great_than</a> <font color="Maroon">c<sub>5</sub></font> implies <font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#NR8" target="_self">is_great_than</a> <font color="Maroon">c<sub>5</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:4_3"/><i><font color="Green">E5</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R1">[=</a> <font color="Maroon">c<sub>3</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a><br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="quantal1.html#E1:4"><i><font color="Green">E3</font></i></a></i>;<br/>
<b>assume </b><a NAME="E3:4_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E4:4_3"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#R1">[=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="quantal1.html#E1:4_3"><i><font color="Green">E5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:4_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#R1">[=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="quantal1.html#E1:4"><i><font color="Green">E3</font></i></a>, <a class="txt" href="quantal1.html#E2:4"><i><font color="Green">E4</font></i></a>, <a class="ref" href="quantal1.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E6:4"/><i><font color="Green">E5</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R1">[=</a> <font color="Maroon">c<sub>4</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a><br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="quantal1.html#E1:4"><i><font color="Green">E3</font></i></a></i>;<br/>
<b>assume </b><a NAME="E8:4"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E9:4"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#R1">[=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="quantal1.html#E6:4"><i><font color="Green">E5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:4"/>
<font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#R1">[=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="quantal1.html#E1:4"><i><font color="Green">E3</font></i></a>, <a class="txt" href="quantal1.html#E2:4"><i><font color="Green">E4</font></i></a>, <a class="ref" href="quantal1.html#T1" target="_self">Th1</a></i>;<br/>


</div>
