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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="funct_1.html#NM1">Function</a>, <font color="Maroon">g</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>



<b>assume </b><a NAME="E1:25"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">B</font>,<font color="Maroon">C</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">f'</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font><b> such that </b><br/><a NAME="E3:25"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> &amp; <font color="Maroon">f'</font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> &amp; <font color="Maroon">f'</font> is <a href="orders_3.html#V5" target="_self">monotone</a> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">g'</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font><b> such that </b><br/><a NAME="E5:25"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g'</font> &amp; <font color="Maroon">g'</font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font> &amp; <font color="Maroon">g'</font> is <a href="orders_3.html#V5" target="_self">monotone</a> )
 <b>by </b><i><a class="ref" href="orders_3.html#D1" target="_self">Def1</a>, </i>;<br/>
<b>consider </b><font color="Maroon">gf</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">A</font>,<font color="Maroon">C</font><b> such that </b><br/><a NAME="E7:25"/><i><font color="Green">E31</font></i>: 
( <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>, , </i>;<br/>
<a NAME="E8:25"/>
<font color="Maroon">gf</font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>hence </b><a NAME="E9:25"/>
<font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">A</font>,<font color="Maroon">C</font>
 <b>by </b><i>, , <a class="txt" href="orders_3.html#E2"><i><font color="Green">Lemma17</font></i></a>, </i>;<br/>


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