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<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="E1:3"/><i><font color="Green">E2</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>, <font color="Maroon">c<sub>4</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>5</sub></font>, <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>;<br/>



<b>assume </b><a NAME="E2:3"/><i><font color="Green">E3</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<b>thus </b><i><font color="Green">E4</font></i>: <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>4</sub></font> = 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>

<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="quantal1.html#E1:3"><i><font color="Green">E2</font></i></a>, <a class="txt" href="quantal1.html#E2:3"><i><font color="Green">E3</font></i></a></i>
;<br/>

<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>4</sub></font> = 
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>

<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="quantal1.html#E1:3"><i><font color="Green">E2</font></i></a>, <a class="txt" href="quantal1.html#E2:3"><i><font color="Green">E3</font></i></a></i>
;<br/>

<a NAME="E5:3"/>
( ( <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R1">[=</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> ) &amp; ( <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R1">[=</a> <font color="Maroon">c<sub>4</sub></font> ) &amp; ( <font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#R1">[=</a> <font color="Maroon">c<sub>6</sub></font> ) &amp; ( <font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#R1">[=</a> <font color="Maroon">c<sub>6</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> ) )
 
<b>by </b><i><a class="ref" href="lattices.html#D3">LATTICES:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:3"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R1">[=</a> <font color="Maroon">c<sub>4</sub></font> iff <font color="Maroon">c<sub>5</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#E2:3"><i><font color="Green">E3</font></i></a>, <a class="txt" href="quantal1.html#E3:3"><i><font color="Green">E4</font></i></a></i>;<br/>


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