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<b>let </b><font color="Maroon">A</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">B</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">C</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">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font>;<br/>





<b>assume </b><a NAME="E1:22"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">f</font> is <a href="orders_3.html#V5" target="_self">monotone</a> &amp; <font color="Maroon">g</font> is <a href="orders_3.html#V5" target="_self">monotone</a> )
 ;<br/>

<b>reconsider </b><font color="Maroon">gf</font> = <font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">A</font>,<font color="Maroon">C</font> ;<br/>
<b>take </b>
<font color="Maroon">gf</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">p2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>;<br/><a NAME="E1:22_1"/><i><font color="Green">E21</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>assume </b><a NAME="E2:22_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">p1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">p2</font>
 ;<br/><b>reconsider </b><font color="Maroon">p1'</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>, <font color="Maroon">p2'</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font> ;<br/><a NAME="E4:22_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E5:22_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">p1'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">p2'</font>
 <b>by </b><i>, , </i>;<br/><b>let </b><font color="Maroon">r1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">C</font>, <font color="Maroon">r2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">C</font>;<br/><b>assume </b><a NAME="E6:22_1"/>
( <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">gf</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">gf</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font> )
 ;<br/><b>hence </b><a NAME="E7:22_1"/>
<font color="Maroon">r1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">r2</font>
 <b>by </b><i>, <a class="ref" href="orders_3.html#D1" target="_self">Def1</a>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:22"/>
( <font color="Maroon">gf</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> &amp; <font color="Maroon">gf</font> is <a href="orders_3.html#V5" target="_self">monotone</a> )
 <b>by </b><i/>;<br/>


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