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<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c3">C</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>



<b>assume </b><a NAME="E1:25"/><i><font color="Green" title="E14">A1</font></i>: 
( <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">f'</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font><b> such that </b><br/><a NAME="E3:25"/><i><font color="Green" title="E15">A2</font></i>: 
( <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f'</font> &amp; <font color="Maroon" title="c6">f'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c6">f'</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> )
 <b>by </b><i><a class="ref" href="orders_3.html#D6" target="_self" title="ORDERS_3:def.6">Def6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">g'</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font><b> such that </b><br/><a NAME="E5:25"/><i><font color="Green" title="E16">A3</font></i>: 
( <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g'</font> &amp; <font color="Maroon" title="c7">g'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C</font> &amp; <font color="Maroon" title="c7">g'</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> )
 <b>by </b><i><a class="txt" href="orders_3.html#E1:25"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="orders_3.html#D6" target="_self" title="ORDERS_3:def.6">Def6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">gf</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font><b> such that </b><br/><a NAME="E7:25"/><i><font color="Green" title="E17">A4</font></i>: 
( <font color="Maroon" title="c8">gf</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">g'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">f'</font> &amp; <font color="Maroon" title="c8">gf</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> )
 <b>by </b><i><a class="txt" href="orders_3.html#E3:25"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="orders_3.html#E5:25"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="orders_3.html#E13"><i><font color="Green" title="E10">Lm3</font></i></a></i>;<br/>
<a NAME="E8:25"/>
<font color="Maroon" title="c8">gf</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<b>hence </b><a NAME="E9:25"/>
<font color="Maroon" title="c5">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font>
 <b>by </b><i><a class="txt" href="orders_3.html#E3:25"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="orders_3.html#E5:25"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="orders_3.html#E7:25"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="orders_3.html#D6" target="_self" title="ORDERS_3:def.6">Def6</a></i>;<br/>


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