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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="nat_lat.html#M2">SubLattice</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</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>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</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="E1:17"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>thus </b><a NAME="E2:17"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>now </b></a><div class="add"><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>;<br/><a NAME="E1:17_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="nat_lat.html#D16">NAT_LAT:def 16</a></i>;<br/><b>then 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="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>assume </b><a NAME="E3:17_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E4:17_1_1"/><b>then </b><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="msualg_7.html#E1:17_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a></i>;<br/><font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>6</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="msualg_7.html#E1:17"><i><font color="Green">E13</font></i></a>, <a class="ref" href="msualg_7.html#T12" target="_self">Th12</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="msualg_7.html#E1:17"><i><font color="Green">E13</font></i></a>, <a class="txt" href="msualg_7.html#E4:17_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="lattices.html#T21">LATTICES:21</a></i>
;<br/><b>hence </b><a NAME="E6:17_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="lattices.html#T21">LATTICES:21</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:17_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a></i>;<br/>


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

<b>thus </b><a NAME="E3:17"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">c<sub>3</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>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_2"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1"><b>now </b></a><div class="add"><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>;<br/><b>assume </b><a NAME="E1:17_2_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then 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> ;<br/><a NAME="E3:17_2_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="msualg_7.html#E1:17_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="msualg_7.html#E1:17_2_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a></i>;<br/><font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>6</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="msualg_7.html#E1:17"><i><font color="Green">E13</font></i></a>, <a class="ref" href="msualg_7.html#T12" target="_self">Th12</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="msualg_7.html#E1:17"><i><font color="Green">E13</font></i></a>, <a class="txt" href="msualg_7.html#E3:17_2_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="lattices.html#T21">LATTICES:21</a></i>
;<br/><b>hence </b><a NAME="E5:17_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="lattices.html#T21">LATTICES:21</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:17_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a></i>;<br/>


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


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