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<b>set </b><font color="Maroon" title="c3">X</font> =  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>;<br/>
<a NAME="E1:58_1_1"/>
 <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font>
 
<b>by </b><i><a class="ref" href="orders_3.html#T11" title="ORDERS_3:th.11">ORDERS_3:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">X</font> =  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">T</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#T28" target="_self" title="YELLOW_1:th.28">Th28</a></i>;<br/>
<b>take </b><font color="Maroon" title="c5">SX</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c4">X</font>;<br/>

<b>let </b><font color="Maroon" title="c6">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:58_1_1_1"/>
<font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">SX</font>
 ;<br/><a NAME="E2:58_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">f'</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font><b> such that </b><br/><a NAME="E4:58_1_1_1"/><i><font color="Green" title="E17">A1</font></i>: 
( <font color="Maroon" title="c7">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c7">f'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font> &amp; <font color="Maroon" title="c7">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" title="ORDERS_3:def.6">ORDERS_3:def 6</a></i>;<br/><b>thus </b><a NAME="E5:58_1_1_1"/>
( <font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</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="txt" href="yellow_1.html#E4:58_1_1_1"><i><font color="Green" title="E17">A1</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:58_1_1"/>
( <font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font> &amp; <font color="Maroon" title="c6">f</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> )
 ;<br/>

<a NAME="E5:58_1_1"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 
<b>by </b><i><a class="ref" href="orders_3.html#D6" title="ORDERS_3:def.6">ORDERS_3:def 6</a></i>;<br/>
<b>hence </b><a NAME="E6:58_1_1"/>
<font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">SX</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/>


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