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<b>let </b><font color="Maroon" title="c1">R</font>, <font color="Maroon" title="c2">S</font>, <font color="Maroon" title="c3">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</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 <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">S</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>,<font color="Maroon" title="c3">T</font><br/> for <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b3">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">f</font> is <a href="waybel_0.html#V5" title="WAYBEL_0:attr.5">antitone</a> holds <br/>(  <a href="waybel24.html#K1" title="WAYBEL24:func.1">Proj</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">a</font> is <a href="waybel_0.html#V5" title="WAYBEL_0:attr.5">antitone</a> &amp;  <a href="waybel24.html#K2" title="WAYBEL24:func.2">Proj</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">b</font> is <a href="waybel_0.html#V5" title="WAYBEL_0:attr.5">antitone</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 <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">S</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>,<font color="Maroon" title="c3">T</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">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Maroon" title="c4">f</font> is <a href="waybel_0.html#V5" title="WAYBEL_0:attr.5">antitone</a> holds <br/>(  <a href="waybel24.html#K1" title="WAYBEL24:func.1">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">a</font> is <a href="waybel_0.html#V5" title="WAYBEL_0:attr.5">antitone</a> &amp;  <a href="waybel24.html#K2" title="WAYBEL24:func.2">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b2">b</font> is <a href="waybel_0.html#V5" title="WAYBEL_0:attr.5">antitone</a> )</span><br/><b>let </b><font color="Maroon" title="c5">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</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">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Maroon" title="c4">f</font> is <a href="waybel_0.html#V5" title="WAYBEL_0:attr.5">antitone</a> holds <br/>(  <a href="waybel24.html#K1" title="WAYBEL24:func.1">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font> is <a href="waybel_0.html#V5" title="WAYBEL_0:attr.5">antitone</a> &amp;  <a href="waybel24.html#K2" title="WAYBEL24:func.2">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">b</font> is <a href="waybel_0.html#V5" title="WAYBEL_0:attr.5">antitone</a> )</span><br/><b>let </b><font color="Maroon" title="c6">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</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="waybel_0.html#V5" title="WAYBEL_0:attr.5">antitone</a> implies (  <a href="waybel24.html#K1" title="WAYBEL24:func.1">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font> is <a href="waybel_0.html#V5" title="WAYBEL_0:attr.5">antitone</a> &amp;  <a href="waybel24.html#K2" title="WAYBEL24:func.2">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">b</font> is <a href="waybel_0.html#V5" title="WAYBEL_0:attr.5">antitone</a> ) )</span><br/>







<b>reconsider </b><font color="Maroon" title="c7">a'</font> = <font color="Maroon" title="c5">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> ;<br/>
<b>set </b><font color="Maroon" title="c8">g</font> =  <a href="waybel24.html#K2" title="WAYBEL24:func.2">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">b</font>;<br/>
<b>set </b><font color="Maroon" title="c9">h</font> =  <a href="waybel24.html#K1" title="WAYBEL24:func.1">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font>;<br/>
<b>assume </b><a NAME="E2:18"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c4">f</font> is <a href="waybel_0.html#V5" title="WAYBEL_0:attr.5">antitone</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="waybel24.html#K1" title="WAYBEL24:func.1">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font> is <a href="waybel_0.html#V5" title="WAYBEL_0:attr.5">antitone</a> &amp;  <a href="waybel24.html#K2" title="WAYBEL24:func.2">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">b</font> is <a href="waybel_0.html#V5" title="WAYBEL_0:attr.5">antitone</a> )</span><br/>

<div><i><font color="Green" title="E9">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">y</font> implies <span class="p1">(<span class="default"><a href="waybel24.html#K1" title="WAYBEL24:func.1">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="waybel24.html#K1" title="WAYBEL24:func.1">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y</font> )</span><br/><b>assume </b><a NAME="E1:18_1"/><i><font color="Green" title="E10">A3</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="waybel24.html#K1" title="WAYBEL24:func.1">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="waybel24.html#K1" title="WAYBEL24:func.1">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y</font></span><br/><a NAME="E2:18_1"/><i><font color="Green" title="E11">A4</font></i>: 
<span class="p1">(<span class="default"><a href="waybel24.html#K1" title="WAYBEL24:func.1">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c10">x</font>
 <b>by </b><i><a class="ref" href="waybel24.html#T7" target="_self" title="WAYBEL24:th.7">Th7</a></i>;<br/><a NAME="E3:18_1"/><i><font color="Green" title="E12">A5</font></i>: 
<span class="p1">(<span class="default"><a href="waybel24.html#K1" title="WAYBEL24:func.1">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c11">y</font>
 <b>by </b><i><a class="ref" href="waybel24.html#T7" target="_self" title="WAYBEL24:th.7">Th7</a></i>;<br/><a NAME="E4:18_1"/>
<font color="Maroon" title="c7">a'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">a'</font>
 ;<br/><a NAME="E5:18_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c10">x</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c11">y</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
 <b>by </b><i><a class="txt" href="#E1:18_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="yellow_3.html#T11" title="YELLOW_3:th.11">YELLOW_3:11</a></i>;<br/><b>hence </b><a NAME="E6:18_1"/>
<span class="p1">(<span class="default"><a href="waybel24.html#K1" title="WAYBEL24:func.1">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="waybel24.html#K1" title="WAYBEL24:func.1">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y</font>
 <b>by </b><i><a class="txt" href="#E2:18"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="#E2:18_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="#E3:18_1"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="waybel_0.html#D5" title="WAYBEL_0:def.5">WAYBEL_0:def 5</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">y</font> implies <span class="p1">(<span class="default"><a href="waybel24.html#K2" title="WAYBEL24:func.2">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">b</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="waybel24.html#K2" title="WAYBEL24:func.2">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">b</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y</font> )</span><br/><b>assume </b><a NAME="E1:18_2"/><i><font color="Green" title="E10">A6</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="waybel24.html#K2" title="WAYBEL24:func.2">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">b</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="waybel24.html#K2" title="WAYBEL24:func.2">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">b</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y</font></span><br/><a NAME="E2:18_2"/><i><font color="Green" title="E11">A7</font></i>: 
<span class="p1">(<span class="default"><a href="waybel24.html#K2" title="WAYBEL24:func.2">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">b</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="ref" href="waybel24.html#T8" target="_self" title="WAYBEL24:th.8">Th8</a></i>;<br/><a NAME="E3:18_2"/><i><font color="Green" title="E12">A8</font></i>: 
<span class="p1">(<span class="default"><a href="waybel24.html#K2" title="WAYBEL24:func.2">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">b</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">y</font>,<font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="ref" href="waybel24.html#T8" target="_self" title="WAYBEL24:th.8">Th8</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">b'</font> = <font color="Maroon" title="c6">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font> ;<br/><a NAME="E5:18_2"/>
<font color="Maroon" title="c12">b'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">b'</font>
 ;<br/><a NAME="E6:18_2"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c12">b'</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c11">y</font>,<font color="Maroon" title="c12">b'</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
 <b>by </b><i><a class="txt" href="#E1:18_2"><i><font color="Green" title="E10">A6</font></i></a>, <a class="ref" href="yellow_3.html#T11" title="YELLOW_3:th.11">YELLOW_3:11</a></i>;<br/><b>hence </b><a NAME="E7:18_2"/>
<span class="p1">(<span class="default"><a href="waybel24.html#K2" title="WAYBEL24:func.2">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">b</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="waybel24.html#K2" title="WAYBEL24:func.2">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">b</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y</font>
 <b>by </b><i><a class="txt" href="#E2:18"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="#E2:18_2"><i><font color="Green" title="E11">A7</font></i></a>, <a class="txt" href="#E3:18_2"><i><font color="Green" title="E12">A8</font></i></a>, <a class="ref" href="waybel_0.html#D5" title="WAYBEL_0:def.5">WAYBEL_0:def 5</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="E5:18"/>
(  <a href="waybel24.html#K1" title="WAYBEL24:func.1">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font> is <a href="waybel_0.html#V5" title="WAYBEL_0:attr.5">antitone</a> &amp;  <a href="waybel24.html#K2" title="WAYBEL24:func.2">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">b</font> is <a href="waybel_0.html#V5" title="WAYBEL_0:attr.5">antitone</a> )
 <b>by </b><i><a class="txt" href="#E3:18"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="waybel_9.html#D1" title="WAYBEL_9:def.1">WAYBEL_9:def 1</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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