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<b>set </b><font color="Maroon" title="c2">A</font> =  <a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font>;<br/>
<b>set </b><font color="Maroon" title="c3">G</font> = the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c4">C</font> = the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span>;<br/>
<b>thus </b><a NAME="E1:39_1_1"/>
the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> is <a href="altcat_1.html#V3" title="ALTCAT_1:attr.3">associative</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D17" title="ALTCAT_1:def.17">ALTCAT_1:def 17</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font> is <a href="altcat_1.html#V12" title="ALTCAT_1:attr.12">with_units</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">i</font>, <font color="Maroon" title="c6">j</font>, <font color="Maroon" title="c7">k</font>, <font color="Maroon" title="c8">l</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D7" title="ALTCAT_1:def.7">ALTCAT_1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c7">k</font> or  not <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c8">l</font> or <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c8">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>3</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c7">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c8">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c8">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c9">i'</font> = <font color="Maroon" title="c5">i</font>, <font color="Maroon" title="c10">j'</font> = <font color="Maroon" title="c6">j</font>, <font color="Maroon" title="c11">k'</font> = <font color="Maroon" title="c7">k</font>, <font color="Maroon" title="c12">l'</font> = <font color="Maroon" title="c8">l</font> as    <a href="orders_3.html#M1" title="ORDERS_3:mode.1">Element</a> of <font color="Maroon" title="c1">P</font> <b>by </b><i><a class="ref" href="orders_3.html#D9" target="_self" title="ORDERS_3:def.9">Def9</a></i>;<br/>
<b>let </b><font color="Maroon" title="c13">f</font>, <font color="Maroon" title="c14">g</font>, <font color="Maroon" title="c15">h</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c13">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font> or  not <font color="Maroon" title="c14">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c7">k</font> or  not <font color="Maroon" title="c15">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c8">l</font> or <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c8">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c15">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c7">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c13">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c8">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c8">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c15">h</font>,<font color="Maroon" title="c14">g</font></span>)</span>,<font color="Maroon" title="c13">f</font> )</span><br/>

<b>assume </b><a NAME="E2:39_1_1_1"/>
( <font color="Maroon" title="c13">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c14">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c15">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c8">l</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">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c8">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c15">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c7">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c13">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c8">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c8">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c15">h</font>,<font color="Maroon" title="c14">g</font></span>)</span>,<font color="Maroon" title="c13">f</font></span><br/>

<a NAME="E3:39_1_1_1"/><b>then </b><i><font color="Green" title="E20">A1</font></i>: 
( <font color="Maroon" title="c13">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="c9">i'</font>,<font color="Maroon" title="c10">j'</font> &amp; <font color="Maroon" title="c14">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="c10">j'</font>,<font color="Maroon" title="c11">k'</font> &amp; <font color="Maroon" title="c15">h</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="c11">k'</font>,<font color="Maroon" title="c12">l'</font> )
 
<b>by </b><i><a class="ref" href="orders_3.html#D9" target="_self" title="ORDERS_3:def.9">Def9</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">f'</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c9">i'</font>,<font color="Maroon" title="c10">j'</font><b> such that </b><br/><a NAME="E5:39_1_1_1"/><i><font color="Green" title="E21">A2</font></i>: 
( <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">f'</font> &amp; <font color="Maroon" title="c16">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="c9">i'</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">j'</font> &amp; <font color="Maroon" title="c16">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="c17">g'</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c10">j'</font>,<font color="Maroon" title="c11">k'</font><b> such that </b><br/><a NAME="E7:39_1_1_1"/><i><font color="Green" title="E22">A3</font></i>: 
( <font color="Maroon" title="c14">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g'</font> &amp; <font color="Maroon" title="c17">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="c10">j'</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">k'</font> &amp; <font color="Maroon" title="c17">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#E3:39_1_1_1"><i><font color="Green" title="E20">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="c18">h'</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c11">k'</font>,<font color="Maroon" title="c12">l'</font><b> such that </b><br/><a NAME="E9:39_1_1_1"/><i><font color="Green" title="E23">A4</font></i>: 
( <font color="Maroon" title="c15">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">h'</font> &amp; <font color="Maroon" title="c18">h'</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="c11">k'</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">l'</font> &amp; <font color="Maroon" title="c18">h'</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#E3:39_1_1_1"><i><font color="Green" title="E20">A1</font></i></a>, <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="c19">f'</font> = <font color="Maroon" title="c13">f</font>, <font color="Maroon" title="c20">g'</font> = <font color="Maroon" title="c14">g</font>, <font color="Maroon" title="c21">h'</font> = <font color="Maroon" title="c15">h</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="txt" href="orders_3.html#E5:39_1_1_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="orders_3.html#E7:39_1_1_1"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="orders_3.html#E9:39_1_1_1"><i><font color="Green" title="E23">A4</font></i></a></i>;<br/>
<a NAME="E11:39_1_1_1"/><i><font color="Green" title="E24">A5</font></i>: 
<font color="Maroon" title="c20">g'</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c19">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="c9">i'</font>,<font color="Maroon" title="c11">k'</font>
 
<b>by </b><i><a class="txt" href="orders_3.html#E3:39_1_1_1"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="orders_3.html#T6" target="_self" title="ORDERS_3:th.6">Th6</a></i>;<br/>
<a NAME="E12:39_1_1_1"/><i><font color="Green" title="E25">A6</font></i>: 
<font color="Maroon" title="c21">h'</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c20">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="c10">j'</font>,<font color="Maroon" title="c12">l'</font>
 
<b>by </b><i><a class="txt" href="orders_3.html#E3:39_1_1_1"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="orders_3.html#T6" target="_self" title="ORDERS_3:th.6">Th6</a></i>;<br/>
<a NAME="E13:39_1_1_1"/><i><font color="Green" title="E26">A7</font></i>: 
the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c8">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> <span class="p1">(<span class="default"><a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> <font color="Maroon" title="c9">i'</font>,<font color="Maroon" title="c11">k'</font></span>)</span>,<span class="p1">(<span class="default"><a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> <font color="Maroon" title="c11">k'</font>,<font color="Maroon" title="c12">l'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="orders_3.html#D9" target="_self" title="ORDERS_3:def.9">Def9</a></i>;<br/>
<a NAME="E14:39_1_1_1"/><i><font color="Green" title="E27">A8</font></i>: 
the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> <span class="p1">(<span class="default"><a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> <font color="Maroon" title="c9">i'</font>,<font color="Maroon" title="c10">j'</font></span>)</span>,<span class="p1">(<span class="default"><a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> <font color="Maroon" title="c10">j'</font>,<font color="Maroon" title="c11">k'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="orders_3.html#D9" target="_self" title="ORDERS_3:def.9">Def9</a></i>;<br/>
<a NAME="E15:39_1_1_1"/><i><font color="Green" title="E28">A9</font></i>: 
the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c8">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> <span class="p1">(<span class="default"><a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> <font color="Maroon" title="c9">i'</font>,<font color="Maroon" title="c10">j'</font></span>)</span>,<span class="p1">(<span class="default"><a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> <font color="Maroon" title="c10">j'</font>,<font color="Maroon" title="c12">l'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="orders_3.html#D9" target="_self" title="ORDERS_3:def.9">Def9</a></i>;<br/>
<a NAME="E16:39_1_1_1"/><i><font color="Green" title="E29">A10</font></i>: 
the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c8">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> <span class="p1">(<span class="default"><a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> <font color="Maroon" title="c10">j'</font>,<font color="Maroon" title="c11">k'</font></span>)</span>,<span class="p1">(<span class="default"><a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> <font color="Maroon" title="c11">k'</font>,<font color="Maroon" title="c12">l'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="orders_3.html#D9" target="_self" title="ORDERS_3:def.9">Def9</a></i>;<br/>
<a NAME="E17:39_1_1_1"/><i><font color="Green" title="E30">A11</font></i>: 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c7">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g'</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c19">f'</font>
 
<b>by </b><i><a class="txt" href="orders_3.html#E3:39_1_1_1"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="orders_3.html#E14:39_1_1_1"><i><font color="Green" title="E27">A8</font></i></a>, <a class="ref" href="altcat_1.html#T13" title="ALTCAT_1:th.13">ALTCAT_1:13</a></i>;<br/>
<a NAME="E18:39_1_1_1"/><i><font color="Green" title="E31">A12</font></i>: 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c8">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c15">h</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c20">g'</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c19">f'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">h'</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g'</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c19">f'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="orders_3.html#E3:39_1_1_1"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="orders_3.html#E11:39_1_1_1"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="orders_3.html#E13:39_1_1_1"><i><font color="Green" title="E26">A7</font></i></a>, <a class="ref" href="altcat_1.html#T13" title="ALTCAT_1:th.13">ALTCAT_1:13</a></i>;<br/>
<a NAME="E19:39_1_1_1"/><i><font color="Green" title="E32">A13</font></i>: 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c8">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c15">h</font>,<font color="Maroon" title="c14">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">h'</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c20">g'</font>
 
<b>by </b><i><a class="txt" href="orders_3.html#E3:39_1_1_1"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="orders_3.html#E16:39_1_1_1"><i><font color="Green" title="E29">A10</font></i></a>, <a class="ref" href="altcat_1.html#T13" title="ALTCAT_1:th.13">ALTCAT_1:13</a></i>;<br/>
<a NAME="E20:39_1_1_1"/>
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c8">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">h'</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c20">g'</font></span>)</span>,<font color="Maroon" title="c19">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">h'</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c20">g'</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c19">f'</font>
 
<b>by </b><i><a class="txt" href="orders_3.html#E3:39_1_1_1"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="orders_3.html#E12:39_1_1_1"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="orders_3.html#E15:39_1_1_1"><i><font color="Green" title="E28">A9</font></i></a>, <a class="ref" href="altcat_1.html#T13" title="ALTCAT_1:th.13">ALTCAT_1:13</a></i>;<br/>
<b>hence </b><a NAME="E21:39_1_1_1"/>
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c8">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c15">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c7">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c13">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c8">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c8">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c15">h</font>,<font color="Maroon" title="c14">g</font></span>)</span>,<font color="Maroon" title="c13">f</font>
 <b>by </b><i><a class="txt" href="orders_3.html#E17:39_1_1_1"><i><font color="Green" title="E30">A11</font></i></a>, <a class="txt" href="orders_3.html#E18:39_1_1_1"><i><font color="Green" title="E31">A12</font></i></a>, <a class="txt" href="orders_3.html#E19:39_1_1_1"><i><font color="Green" title="E32">A13</font></i></a>, <a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</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>thus </b><a NAME="E2:39_1_1"/>
the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> is <a href="altcat_1.html#V5" title="ALTCAT_1:attr.5">with_left_units</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D18" title="ALTCAT_1:def.18">ALTCAT_1:def 18</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> is <a href="altcat_1.html#V4" title="ALTCAT_1:attr.4">with_right_units</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D9" title="ALTCAT_1:def.9">ALTCAT_1:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c5">j</font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span><br/> for <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c5">j</font> or <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>3</sub></font> ) ) )</span><br/>

<b>reconsider </b><font color="Maroon" title="c6">j'</font> = <font color="Maroon" title="c5">j</font> as    <a href="orders_3.html#M1" title="ORDERS_3:mode.1">Element</a> of <font color="Maroon" title="c1">P</font> <b>by </b><i><a class="ref" href="orders_3.html#D9" target="_self" title="ORDERS_3:def.9">Def9</a></i>;<br/>
<b>take </b><font color="Maroon" title="c7">e</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">j'</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">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c5">j</font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span><br/> for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c5">j</font> or <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">e</font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) )</span><br/>

<a NAME="E2:39_1_1_2"/>
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> <font color="Maroon" title="c6">j'</font>,<font color="Maroon" title="c6">j'</font>
 
<b>by </b><i><a class="ref" href="orders_3.html#D9" target="_self" title="ORDERS_3:def.9">Def9</a></i>;<br/>
<b>hence </b><a NAME="E3:39_1_1_2"/>
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c5">j</font>
 <b>by </b><i><a class="ref" href="orders_3.html#T7" target="_self" title="ORDERS_3:th.7">Th7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span><br/> for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c5">j</font> or <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">e</font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> )</span><br/>

<a NAME="E4:39_1_1_2"/><i><font color="Green" title="E20">A14</font></i>: 
<font color="Maroon" title="c7">e</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="c6">j'</font>,<font color="Maroon" title="c6">j'</font>
 
<b>by </b><i><a class="ref" href="orders_3.html#T7" target="_self" title="ORDERS_3:th.7">Th7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">e'</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c6">j'</font>,<font color="Maroon" title="c6">j'</font><b> such that </b><br/><a NAME="E6:39_1_1_2"/><i><font color="Green" title="E21">A15</font></i>: 
( <font color="Maroon" title="c7">e</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">e'</font> &amp; <font color="Maroon" title="c8">e'</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="c6">j'</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">j'</font> &amp; <font color="Maroon" title="c8">e'</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>let </b><font color="Maroon" title="c9">i</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c5">j</font> or <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">e</font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><b>let </b><font color="Maroon" title="c10">f</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c10">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c5">j</font> or <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">e</font>,<font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> )</span><br/>



<b>reconsider </b><font color="Maroon" title="c11">i'</font> = <font color="Maroon" title="c9">i</font> as    <a href="orders_3.html#M1" title="ORDERS_3:mode.1">Element</a> of <font color="Maroon" title="c1">P</font> <b>by </b><i><a class="ref" href="orders_3.html#D9" target="_self" title="ORDERS_3:def.9">Def9</a></i>;<br/>
<b>assume </b><a NAME="E8:39_1_1_2"/>
<font color="Maroon" title="c10">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c5">j</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">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">e</font>,<font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font></span><br/>

<a NAME="E9:39_1_1_2"/><b>then </b><i><font color="Green" title="E22">A16</font></i>: 
<font color="Maroon" title="c10">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="c11">i'</font>,<font color="Maroon" title="c6">j'</font>
 
<b>by </b><i><a class="ref" href="orders_3.html#D9" target="_self" title="ORDERS_3:def.9">Def9</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">f'</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c11">i'</font>,<font color="Maroon" title="c6">j'</font><b> such that </b><br/><a NAME="E11:39_1_1_2"/><i><font color="Green" title="E23">A17</font></i>: 
( <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f'</font> &amp; <font color="Maroon" title="c12">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="c11">i'</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">j'</font> &amp; <font color="Maroon" title="c12">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/>
<a NAME="E12:39_1_1_2"/><i><font color="Green" title="E24">A18</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">f'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">j'</font>
 
;<br/>
<a NAME="E13:39_1_1_2"/><i><font color="Green" title="E25">A19</font></i>: 
the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> <span class="p1">(<span class="default"><a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> <font color="Maroon" title="c11">i'</font>,<font color="Maroon" title="c6">j'</font></span>)</span>,<span class="p1">(<span class="default"><a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> <font color="Maroon" title="c6">j'</font>,<font color="Maroon" title="c6">j'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="orders_3.html#D9" target="_self" title="ORDERS_3:def.9">Def9</a></i>;<br/>
<a NAME="E14:39_1_1_2"/>
<font color="Maroon" title="c8">e'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font>
 
<b>by </b><i><a class="txt" href="orders_3.html#E6:39_1_1_2"><i><font color="Green" title="E21">A15</font></i></a>, <a class="txt" href="orders_3.html#E11:39_1_1_2"><i><font color="Green" title="E23">A17</font></i></a>, <a class="txt" href="orders_3.html#E12:39_1_1_2"><i><font color="Green" title="E24">A18</font></i></a>, <a class="ref" href="relat_1.html#T79" title="RELAT_1:th.79">RELAT_1:79</a></i>;<br/>
<b>hence </b><a NAME="E15:39_1_1_2"/>
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">e</font>,<font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font>
 <b>by </b><i><a class="txt" href="orders_3.html#E4:39_1_1_2"><i><font color="Green" title="E20">A14</font></i></a>, <a class="txt" href="orders_3.html#E6:39_1_1_2"><i><font color="Green" title="E21">A15</font></i></a>, <a class="txt" href="orders_3.html#E9:39_1_1_2"><i><font color="Green" title="E22">A16</font></i></a>, <a class="txt" href="orders_3.html#E11:39_1_1_2"><i><font color="Green" title="E23">A17</font></i></a>, <a class="txt" href="orders_3.html#E13:39_1_1_2"><i><font color="Green" title="E25">A19</font></i></a>, <a class="ref" href="altcat_1.html#T13" title="ALTCAT_1:th.13">ALTCAT_1:13</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>thus </b><a NAME="E3:39_1_1"/>
the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> is <a href="altcat_1.html#V4" title="ALTCAT_1:attr.4">with_right_units</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">i</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D8" title="ALTCAT_1:def.8">ALTCAT_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c5">i</font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span><br/> for <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<font color="Olive">b<sub>2</sub></font> or <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c5">i</font>,<font color="Olive">b<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>3</sub></font> ) ) )</span><br/>

<b>reconsider </b><font color="Maroon" title="c6">i'</font> = <font color="Maroon" title="c5">i</font> as    <a href="orders_3.html#M1" title="ORDERS_3:mode.1">Element</a> of <font color="Maroon" title="c1">P</font> <b>by </b><i><a class="ref" href="orders_3.html#D9" target="_self" title="ORDERS_3:def.9">Def9</a></i>;<br/>
<b>take </b><font color="Maroon" title="c7">e</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">i'</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">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c5">i</font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span><br/> for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<font color="Olive">b<sub>1</sub></font> or <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c5">i</font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c7">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) )</span><br/>

<a NAME="E2:39_1_1_3"/>
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> <font color="Maroon" title="c6">i'</font>,<font color="Maroon" title="c6">i'</font>
 
<b>by </b><i><a class="ref" href="orders_3.html#D9" target="_self" title="ORDERS_3:def.9">Def9</a></i>;<br/>
<b>hence </b><a NAME="E3:39_1_1_3"/>
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="ref" href="orders_3.html#T7" target="_self" title="ORDERS_3:th.7">Th7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span><br/> for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<font color="Olive">b<sub>1</sub></font> or <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c5">i</font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c7">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> )</span><br/>

<a NAME="E4:39_1_1_3"/><i><font color="Green" title="E20">A20</font></i>: 
<font color="Maroon" title="c7">e</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="c6">i'</font>,<font color="Maroon" title="c6">i'</font>
 
<b>by </b><i><a class="ref" href="orders_3.html#T7" target="_self" title="ORDERS_3:th.7">Th7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">e'</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c6">i'</font>,<font color="Maroon" title="c6">i'</font><b> such that </b><br/><a NAME="E6:39_1_1_3"/><i><font color="Green" title="E21">A21</font></i>: 
( <font color="Maroon" title="c7">e</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">e'</font> &amp; <font color="Maroon" title="c8">e'</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="c6">i'</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">i'</font> &amp; <font color="Maroon" title="c8">e'</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>let </b><font color="Maroon" title="c9">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c9">j</font> or <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c7">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><b>let </b><font color="Maroon" title="c10">f</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c10">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c9">j</font> or <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c7">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> )</span><br/>



<b>reconsider </b><font color="Maroon" title="c11">j'</font> = <font color="Maroon" title="c9">j</font> as    <a href="orders_3.html#M1" title="ORDERS_3:mode.1">Element</a> of <font color="Maroon" title="c1">P</font> <b>by </b><i><a class="ref" href="orders_3.html#D9" target="_self" title="ORDERS_3:def.9">Def9</a></i>;<br/>
<b>assume </b><a NAME="E8:39_1_1_3"/>
<font color="Maroon" title="c10">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c9">j</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">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c7">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font></span><br/>

<a NAME="E9:39_1_1_3"/><b>then </b><i><font color="Green" title="E22">A22</font></i>: 
<font color="Maroon" title="c10">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="c6">i'</font>,<font color="Maroon" title="c11">j'</font>
 
<b>by </b><i><a class="ref" href="orders_3.html#D9" target="_self" title="ORDERS_3:def.9">Def9</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">f'</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c6">i'</font>,<font color="Maroon" title="c11">j'</font><b> such that </b><br/><a NAME="E11:39_1_1_3"/><i><font color="Green" title="E23">A23</font></i>: 
( <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f'</font> &amp; <font color="Maroon" title="c12">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="c6">i'</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">j'</font> &amp; <font color="Maroon" title="c12">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/>
<a NAME="E12:39_1_1_3"/><i><font color="Green" title="E24">A24</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">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="c6">i'</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/>
<a NAME="E13:39_1_1_3"/><i><font color="Green" title="E25">A25</font></i>: 
the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> <span class="p1">(<span class="default"><a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> <font color="Maroon" title="c6">i'</font>,<font color="Maroon" title="c6">i'</font></span>)</span>,<span class="p1">(<span class="default"><a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> <font color="Maroon" title="c6">i'</font>,<font color="Maroon" title="c11">j'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="orders_3.html#D9" target="_self" title="ORDERS_3:def.9">Def9</a></i>;<br/>
<a NAME="E14:39_1_1_3"/>
<font color="Maroon" title="c12">f'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">e'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font>
 
<b>by </b><i><a class="txt" href="orders_3.html#E6:39_1_1_3"><i><font color="Green" title="E21">A21</font></i></a>, <a class="txt" href="orders_3.html#E11:39_1_1_3"><i><font color="Green" title="E23">A23</font></i></a>, <a class="txt" href="orders_3.html#E12:39_1_1_3"><i><font color="Green" title="E24">A24</font></i></a>, <a class="ref" href="relat_1.html#T78" title="RELAT_1:th.78">RELAT_1:78</a></i>;<br/>
<b>hence </b><a NAME="E15:39_1_1_3"/>
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c7">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font>
 <b>by </b><i><a class="txt" href="orders_3.html#E4:39_1_1_3"><i><font color="Green" title="E20">A20</font></i></a>, <a class="txt" href="orders_3.html#E6:39_1_1_3"><i><font color="Green" title="E21">A21</font></i></a>, <a class="txt" href="orders_3.html#E9:39_1_1_3"><i><font color="Green" title="E22">A22</font></i></a>, <a class="txt" href="orders_3.html#E11:39_1_1_3"><i><font color="Green" title="E23">A23</font></i></a>, <a class="txt" href="orders_3.html#E13:39_1_1_3"><i><font color="Green" title="E25">A25</font></i></a>, <a class="ref" href="altcat_1.html#T13" title="ALTCAT_1:th.13">ALTCAT_1:13</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>
