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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#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</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">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><br/> for <font color="Olive" title="b2">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>  st <font color="Olive" title="b1">f</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> &amp; <font color="Olive" title="b2">g</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> holds <br/> ex <font color="Olive" title="b3">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> st <br/>( <font color="Olive" title="b3">gf</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b3">gf</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> )</span><br/><b>let </b><font color="Maroon" title="c4">f</font> be   <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>; <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">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>  st <font color="Maroon" title="c4">f</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> &amp; <font color="Olive" title="b1">g</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> holds <br/> ex <font color="Olive" title="b2">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> st <br/>( <font color="Olive" title="b2">gf</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Olive" title="b2">gf</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> )</span><br/><b>let </b><font color="Maroon" title="c5">g</font> be   <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">f</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> &amp; <font color="Maroon" title="c5">g</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> implies  ex <font color="Olive" title="b1">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> st <br/>( <font color="Olive" title="b1">gf</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Olive" title="b1">gf</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> ) )</span><br/>





<b>assume </b><a NAME="E1:21"/><i><font color="Green" title="E11">A1</font></i>: 
( <font color="Maroon" title="c4">f</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> &amp; <font color="Maroon" title="c5">g</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">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> st <br/>( <font color="Olive" title="b1">gf</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Olive" title="b1">gf</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c6">gf</font> = <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font> as   <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> ;<br/>
<b>take </b>
<font color="Maroon" title="c6">gf</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">gf</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c6">gf</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">p1</font>, <font color="Maroon" title="c8">p2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">p1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">p2</font> implies for <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">C</font>  st <font color="Olive" title="b1">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">gf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Olive" title="b2">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">gf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p2</font> holds <br/><font color="Olive" title="b1">r1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">r2</font> )</span><br/><a NAME="E1:21_1"/><i><font color="Green" title="E12">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>assume </b><a NAME="E2:21_1"/><i><font color="Green" title="E13">A3</font></i>: 
<font color="Maroon" title="c7">p1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">p2</font>
 ; <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">r1</font>, <font color="Olive" title="b2">r2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">C</font>  st <font color="Olive" title="b1">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">gf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Olive" title="b2">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">gf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p2</font> holds <br/><font color="Olive" title="b1">r1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">r2</font></span><br/><b>reconsider </b><font color="Maroon" title="c9">p1'</font> = <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p1</font>, <font color="Maroon" title="c10">p2'</font> = <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">B</font> ;<br/><a NAME="E4:21_1"/><i><font color="Green" title="E14">A4</font></i>: 
( <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p2</font> )
 <b>by </b><i><a class="txt" href="orders_3.html#E1:21_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E5:21_1"/><i><font color="Green" title="E15">A5</font></i>: 
<font color="Maroon" title="c9">p1'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">p2'</font>
 <b>by </b><i><a class="txt" href="orders_3.html#E1:21"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="orders_3.html#E2:21_1"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="orders_3.html#D5" target="_self" title="ORDERS_3:def.5">Def5</a></i>;<br/><b>let </b><font color="Maroon" title="c11">r1</font>, <font color="Maroon" title="c12">r2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">gf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c12">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">gf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p2</font> implies <font color="Maroon" title="c11">r1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c12">r2</font> )</span><br/><b>assume </b><a NAME="E6:21_1"/>
( <font color="Maroon" title="c11">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">gf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c12">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">gf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">r1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c12">r2</font></span><br/><b>hence </b><a NAME="E7:21_1"/>
<font color="Maroon" title="c11">r1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c12">r2</font>
 <b>by </b><i><a class="txt" href="orders_3.html#E1:21"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="orders_3.html#E4:21_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="orders_3.html#E5:21_1"><i><font color="Green" title="E15">A5</font></i></a>, <a class="ref" href="orders_3.html#D5" target="_self" title="ORDERS_3:def.5">Def5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:21"/>
( <font color="Maroon" title="c6">gf</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c6">gf</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#D5" target="_self" title="ORDERS_3:def.5">Def5</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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