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<b>let </b><font color="Maroon">L1</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">L2</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">L3</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">g1</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L1</font>,<font color="Maroon">L2</font>;<br/><b>let </b><font color="Maroon">g2</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L2</font>,<font color="Maroon">L3</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:17"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">g1</font> is <a href="orders_3.html#V5">monotone</a>
 <b>and </b><br/><a NAME="E2:17"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">g2</font> is <a href="orders_3.html#V5">monotone</a>
 ;<br/>

<b>let </b><font color="Maroon">s1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>, <font color="Maroon">s2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_3.html#D5">ORDERS_3:def 5</a><br/>

<b>assume </b><a NAME="E3:17"/>
<font color="Maroon">s1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s2</font>
 ;<br/>

<a NAME="E4:17"/><b>then </b>
<font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s2</font>
 
<b>by </b><i>, <a class="ref" href="orders_3.html#D5">ORDERS_3:def 5</a></i>;<br/>
<a NAME="E5:17"/><b>then </b>
<font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s1</font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="orders_3.html#D5">ORDERS_3:def 5</a></i>;<br/>
<a NAME="E6:17"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/>
<b>hence </b><a NAME="E7:17"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L3</font>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> or <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/>


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