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<b>let </b><font color="Maroon" title="c1">P</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="orders_3.html#NM1" title="ORDERS_3:NM.1">POSet_set</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="orders_3.html#M1" title="ORDERS_3:mode.1">Element</a> of <font color="Maroon" title="c1">P</font> holds   <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" title="ORDERS_3:func.2">Carr</a> <font color="Maroon" title="c1">P</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be    <a href="orders_3.html#M1" title="ORDERS_3:mode.1">Element</a> of <font color="Maroon" title="c1">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" title="ORDERS_3:func.2">Carr</a> <font color="Maroon" title="c1">P</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">x</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">A</font>,<font color="Maroon" title="c3">B</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" title="ORDERS_3:func.2">Carr</a> <font color="Maroon" title="c1">P</font></span>)</span> )</span><br/>







<b>assume </b><a NAME="E1:31"/>
<font color="Maroon" title="c4">x</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">A</font>,<font color="Maroon" title="c3">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" title="ORDERS_3:func.2">Carr</a> <font color="Maroon" title="c1">P</font></span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c5">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font><b> such that </b><br/><a NAME="E3:31"/><i><font color="Green" title="E17">A1</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> &amp; <font color="Maroon" title="c5">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">A</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c5">g</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>reconsider </b><font color="Maroon" title="c6">CA</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c7">CB</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">B</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="orders_3.html#K2" title="ORDERS_3:func.2">Carr</a> <font color="Maroon" title="c1">P</font> <b>by </b><i><a class="ref" href="orders_3.html#D7" target="_self" title="ORDERS_3:def.7">Def7</a></i>;<br/>
<a NAME="E5:31"/>
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c6">CA</font>,<font color="Maroon" title="c7">CB</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" title="ORDERS_3:func.2">Carr</a> <font color="Maroon" title="c1">P</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ens_1.html#T2" title="ENS_1:th.2">ENS_1:2</a></i>;<br/>
<b>hence </b><a NAME="E6:31"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" title="ORDERS_3:func.2">Carr</a> <font color="Maroon" title="c1">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:31"><i><font color="Green" title="E17">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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