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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:61"/><i><font color="Green">E38</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> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>2</sub></font> iff <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>given </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E2:61"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E3:61"/><i><font color="Green">E40</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="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font>
 <b>and </b><br/><a NAME="E4:61"/><i><font color="Green">E41</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="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> ) holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D7" target="_self">YELLOW_0:def 7</a><br/>

<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D7" target="_self">YELLOW_0:def 7</a><br/>

<b>thus </b><a NAME="E5:61"/>
<font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E1:61"><i><font color="Green">E38</font></i></a>, <a class="txt" href="yellow_0.html#E2:61"><i><font color="Green">E39</font></i></a></i>;<br/>

<div><b>hereby </b>
<div class="add"><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>1</sub></font>;<br/><b>assume </b><a NAME="E1:61_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:61_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E1:61"><i><font color="Green">E38</font></i></a></i>;<br/><b>hence </b><a NAME="E3:61_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E3:61"><i><font color="Green">E40</font></i></a></i>;<br/></div>
<b>end;</b></div>

<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>1</sub></font>;<br/>

<b>assume </b><a NAME="E7:61"/>
<font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E8:61"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="yellow_0.html#E1:61"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>assume </b><a NAME="E9:61"/><i><font color="Green">E43</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="Maroon">c<sub>3</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2"><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="E1:61_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:61_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E1:61"><i><font color="Green">E38</font></i></a></i>;<br/><b>hence </b><a NAME="E3:61_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E9:61"><i><font color="Green">E43</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:61"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E4:61"><i><font color="Green">E41</font></i></a>, <a class="txt" href="yellow_0.html#E8:61"><i><font color="Green">E42</font></i></a></i>;<br/>


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