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<span class="kw">set </span><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/>
<span class="kw">set </span><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/>
<span class="kw">set </span><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/>
<span class="kw">thus </span><a NAME="E1:37_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> 
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="altcat_1.html#D15" onmouseover="rs('altcat_1/D15')" onmouseout="rh()">ALTCAT_1:def 15</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="37_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="altcat_1.html#D5" onmouseover="rs('altcat_1/D5')" onmouseout="rh()">ALTCAT_1:def 5</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#M2" title="HIDDEN:mode.2">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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/><span class="kw">let </span><font color="Maroon" title="c9">f</font>, <font color="Maroon" title="c10">g</font>, <font color="Maroon" title="c11">h</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">f</font> <a href="tarski.html#R2" title="TARSKI: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="c10">g</font> <a href="tarski.html#R2" title="TARSKI: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="c11">h</font> <a href="tarski.html#R2" title="TARSKI: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="c11">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="c10">g</font>,<font color="Maroon" title="c9">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="c11">h</font>,<font color="Maroon" title="c10">g</font>)</span>)</span>,<font color="Maroon" title="c9">f</font>) )</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c12">i9</font> = <font color="Maroon" title="c5">i</font>, <font color="Maroon" title="c13">j9</font> = <font color="Maroon" title="c6">j</font>, <font color="Maroon" title="c14">k9</font> = <font color="Maroon" title="c7">k</font>, <font color="Maroon" title="c15">l9</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> <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_3.html#D9" target="_self" onmouseover="rs('orders_3/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:37_1_1_1"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="tarski.html#R2" title="TARSKI: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>)
 <span class="kw">and </span><br/><a NAME="E3:37_1_1_1"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<font color="Maroon" title="c10">g</font> <a href="tarski.html#R2" title="TARSKI: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>)
 <span class="kw">and </span><br/><a NAME="E4:37_1_1_1"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
<font color="Maroon" title="c11">h</font> <a href="tarski.html#R2" title="TARSKI: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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c11">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="c10">g</font>,<font color="Maroon" title="c9">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="c11">h</font>,<font color="Maroon" title="c10">g</font>)</span>)</span>,<font color="Maroon" title="c9">f</font>)</span><br/>

<a NAME="E5:37_1_1_1"/><span class="lab"><font color="Green" title="E17">A4</font></span>: 
<font color="Maroon" title="c10">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> (<font color="Maroon" title="c13">j9</font>,<font color="Maroon" title="c14">k9</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_1_1_1"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="orders_3.html#D9" target="_self" onmouseover="rs('orders_3/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<a NAME="E6:37_1_1_1"/><span class="lab"><font color="Green" title="E18">A5</font></span>: 
<font color="Maroon" title="c11">h</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> (<font color="Maroon" title="c14">k9</font>,<font color="Maroon" title="c15">l9</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_1_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="orders_3.html#D9" target="_self" onmouseover="rs('orders_3/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<a NAME="E7:37_1_1_1"/><span class="lab"><font color="Green" title="E19">A6</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> (<font color="Maroon" title="c12">i9</font>,<font color="Maroon" title="c13">j9</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1_1_1"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="orders_3.html#D9" target="_self" onmouseover="rs('orders_3/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">f9</font> = <font color="Maroon" title="c9">f</font>, <font color="Maroon" title="c17">g9</font> = <font color="Maroon" title="c10">g</font>, <font color="Maroon" title="c18">h9</font> = <font color="Maroon" title="c11">h</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37_1_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E6:37_1_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a></span>;<br/>
<a NAME="E9:37_1_1_1"/><span class="lab"><font color="Green" title="E20">A7</font></span>: 
 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="c12">i9</font>,<font color="Maroon" title="c13">j9</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="c13">j9</font>,<font color="Maroon" title="c15">l9</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_3.html#D9" target="_self" onmouseover="rs('orders_3/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<a NAME="E10:37_1_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> <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="c13">j9</font>,<font color="Maroon" title="c14">k9</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="c14">k9</font>,<font color="Maroon" title="c15">l9</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_3.html#D9" target="_self" onmouseover="rs('orders_3/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<a NAME="E11:37_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A8</font></span>: 
<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="c11">h</font>,<font color="Maroon" title="c10">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">h9</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c17">g9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37_1_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E6:37_1_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="ref" href="altcat_1.html#T11" onmouseover="rs('altcat_1/T11')" onmouseout="rh()">ALTCAT_1:11</a></span>;<br/>
<a NAME="E12:37_1_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> <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="c12">i9</font>,<font color="Maroon" title="c13">j9</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="c13">j9</font>,<font color="Maroon" title="c14">k9</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_3.html#D9" target="_self" onmouseover="rs('orders_3/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<a NAME="E13:37_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A9</font></span>: 
<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="c10">g</font>,<font color="Maroon" title="c9">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g9</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c16">f9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:37_1_1_1"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E5:37_1_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="altcat_1.html#T11" onmouseover="rs('altcat_1/T11')" onmouseout="rh()">ALTCAT_1:11</a></span>;<br/>
<a NAME="E14:37_1_1_1"/>
<font color="Maroon" title="c18">h9</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c17">g9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> (<font color="Maroon" title="c13">j9</font>,<font color="Maroon" title="c15">l9</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37_1_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E6:37_1_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="ref" href="orders_3.html#T6" target="_self" onmouseover="rs('orders_3/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<a NAME="E15:37_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A10</font></span>: 
<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="c18">h9</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c17">g9</font></span>)</span>,<font color="Maroon" title="c16">f9</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">h9</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c17">g9</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c16">f9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:37_1_1_1"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E9:37_1_1_1"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="altcat_1.html#T11" onmouseover="rs('altcat_1/T11')" onmouseout="rh()">ALTCAT_1:11</a></span>;<br/>
<a NAME="E16:37_1_1_1"/><span class="lab"><font color="Green" title="E24">A11</font></span>: 
 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="c12">i9</font>,<font color="Maroon" title="c14">k9</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="c14">k9</font>,<font color="Maroon" title="c15">l9</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_3.html#D9" target="_self" onmouseover="rs('orders_3/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<a NAME="E17:37_1_1_1"/>
<font color="Maroon" title="c17">g9</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c16">f9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> (<font color="Maroon" title="c12">i9</font>,<font color="Maroon" title="c14">k9</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:37_1_1_1"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E5:37_1_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="orders_3.html#T6" target="_self" onmouseover="rs('orders_3/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<a NAME="E18:37_1_1_1"/><span class="kw">then </span>
<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="c11">h</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">g9</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c16">f9</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">h9</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">g9</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c16">f9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E16:37_1_1_1"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="ref" href="altcat_1.html#T11" onmouseover="rs('altcat_1/T11')" onmouseout="rh()">ALTCAT_1:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:37_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="c11">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="c10">g</font>,<font color="Maroon" title="c9">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="c11">h</font>,<font color="Maroon" title="c10">g</font>)</span>)</span>,<font color="Maroon" title="c9">f</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:37_1_1_1"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="txt" href="#E11:37_1_1_1"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="txt" href="#E15:37_1_1_1"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="ref" href="relat_1.html#T36" onmouseover="rs('relat_1/T36')" onmouseout="rh()">RELAT_1:36</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E2:37_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> 
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="altcat_1.html#D16" onmouseover="rs('altcat_1/D16')" onmouseout="rh()">ALTCAT_1:def 16</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="37_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="altcat_1.html#D7" onmouseover="rs('altcat_1/D7')" onmouseout="rh()">ALTCAT_1:def 7</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R2" title="TARSKI: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#M2" title="HIDDEN:mode.2">set</a>   holds <br/>(  not <font color="Olive">b<sub>3</sub></font> <a href="tarski.html#R2" title="TARSKI: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/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">j9</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> <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_3.html#D9" target="_self" onmouseover="rs('orders_3/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<span class="kw">take </span><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">j9</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">e</font> <a href="tarski.html#R2" title="TARSKI: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#M2" title="HIDDEN:mode.2">set</a>   holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R2" title="TARSKI: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="E2:37_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">j9</font>,<font color="Maroon" title="c6">j9</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_3.html#D9" target="_self" onmouseover="rs('orders_3/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:37_1_1_2"/>
<font color="Maroon" title="c7">e</font> <a href="tarski.html#R2" title="TARSKI: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>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_3.html#T7" target="_self" onmouseover="rs('orders_3/T7')" onmouseout="rh()">Th7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#M2" title="HIDDEN:mode.2">set</a>   holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R2" title="TARSKI: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/>

<span class="kw">let </span><font color="Maroon" title="c8">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R2" title="TARSKI: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="c8">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="c8">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/><span class="kw">let </span><font color="Maroon" title="c9">f</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">f</font> <a href="tarski.html#R2" title="TARSKI: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="c8">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="c8">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="c9">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> )</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c10">i9</font> = <font color="Maroon" title="c8">i</font> as    <a href="orders_3.html#M1" title="ORDERS_3:mode.1">Element</a> of <font color="Maroon" title="c1">P</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_3.html#D9" target="_self" onmouseover="rs('orders_3/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<a NAME="E5:37_1_1_2"/><span class="lab"><font color="Green" title="E14">A12</font></span>: 
 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="c8">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="c10">i9</font>,<font color="Maroon" title="c6">j9</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">j9</font>,<font color="Maroon" title="c6">j9</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_3.html#D9" target="_self" onmouseover="rs('orders_3/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:37_1_1_2"/>
<font color="Maroon" title="c9">f</font> <a href="tarski.html#R2" title="TARSKI: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="c8">i</font>,<font color="Maroon" title="c5">j</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c8">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="c9">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font></span><br/>

<a NAME="E7:37_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A13</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> (<font color="Maroon" title="c10">i9</font>,<font color="Maroon" title="c6">j9</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_3.html#D9" target="_self" onmouseover="rs('orders_3/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">f9</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c10">i9</font>,<font color="Maroon" title="c6">j9</font><span class="kw"> such that </span><br/><a NAME="E9:37_1_1_2"/><span class="lab"><font color="Green" title="E16">A14</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f9</font>
 <span class="kw">and </span><br/><a NAME="E10:37_1_1_2"/>
<font color="Maroon" title="c11">f9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">i9</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">j9</font>)
 <span class="kw">and </span><br/><a NAME="E11:37_1_1_2"/>
<font color="Maroon" title="c11">f9</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_3.html#D6" target="_self" onmouseover="rs('orders_3/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<a NAME="E12:37_1_1_2"/><span class="lab"><font color="Green" title="E17">A15</font></span>: 
<font color="Maroon" title="c7">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> (<font color="Maroon" title="c6">j9</font>,<font color="Maroon" title="c6">j9</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_3.html#T7" target="_self" onmouseover="rs('orders_3/T7')" onmouseout="rh()">Th7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">e9</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c6">j9</font>,<font color="Maroon" title="c6">j9</font><span class="kw"> such that </span><br/><a NAME="E14:37_1_1_2"/><span class="lab"><font color="Green" title="E18">A16</font></span>: 
<font color="Maroon" title="c7">e</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c12">e9</font>
 <span class="kw">and </span><br/><a NAME="E15:37_1_1_2"/>
<font color="Maroon" title="c12">e9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">j9</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">j9</font>)
 <span class="kw">and </span><br/><a NAME="E16:37_1_1_2"/>
<font color="Maroon" title="c12">e9</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_3.html#D6" target="_self" onmouseover="rs('orders_3/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<a NAME="E17:37_1_1_2"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">f9</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">j9</font>
 
;<br/>
<a NAME="E18:37_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">e9</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:37_1_1_2"><span class="lab"><font color="Green" title="E18">A16</font></span></a>, <a class="txt" href="#E9:37_1_1_2"><span class="lab"><font color="Green" title="E16">A14</font></span></a>, <a class="ref" href="relat_1.html#T53" onmouseover="rs('relat_1/T53')" onmouseout="rh()">RELAT_1:53</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:37_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="c8">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="c9">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:37_1_1_2"><span class="lab"><font color="Green" title="E17">A15</font></span></a>, <a class="txt" href="#E14:37_1_1_2"><span class="lab"><font color="Green" title="E18">A16</font></span></a>, <a class="txt" href="#E7:37_1_1_2"><span class="lab"><font color="Green" title="E15">A13</font></span></a>, <a class="txt" href="#E9:37_1_1_2"><span class="lab"><font color="Green" title="E16">A14</font></span></a>, <a class="txt" href="#E5:37_1_1_2"><span class="lab"><font color="Green" title="E14">A12</font></span></a>, <a class="ref" href="altcat_1.html#T11" onmouseover="rs('altcat_1/T11')" onmouseout="rh()">ALTCAT_1:11</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E3:37_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="37_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="altcat_1.html#D6" onmouseover="rs('altcat_1/D6')" onmouseout="rh()">ALTCAT_1:def 6</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R2" title="TARSKI: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#M2" title="HIDDEN:mode.2">set</a>   holds <br/>(  not <font color="Olive">b<sub>3</sub></font> <a href="tarski.html#R2" title="TARSKI: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/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">i9</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> <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_3.html#D9" target="_self" onmouseover="rs('orders_3/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<span class="kw">take </span><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">i9</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">e</font> <a href="tarski.html#R2" title="TARSKI: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#M2" title="HIDDEN:mode.2">set</a>   holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R2" title="TARSKI: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="E2:37_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">i9</font>,<font color="Maroon" title="c6">i9</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_3.html#D9" target="_self" onmouseover="rs('orders_3/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:37_1_1_3"/>
<font color="Maroon" title="c7">e</font> <a href="tarski.html#R2" title="TARSKI: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>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_3.html#T7" target="_self" onmouseover="rs('orders_3/T7')" onmouseout="rh()">Th7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#M2" title="HIDDEN:mode.2">set</a>   holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R2" title="TARSKI: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/>

<span class="kw">let </span><font color="Maroon" title="c8">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R2" title="TARSKI: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="c8">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="c8">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/><span class="kw">let </span><font color="Maroon" title="c9">f</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">f</font> <a href="tarski.html#R2" title="TARSKI: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="c8">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="c8">j</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">f</font>,<font color="Maroon" title="c7">e</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> )</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c10">j9</font> = <font color="Maroon" title="c8">j</font> as    <a href="orders_3.html#M1" title="ORDERS_3:mode.1">Element</a> of <font color="Maroon" title="c1">P</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_3.html#D9" target="_self" onmouseover="rs('orders_3/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<a NAME="E5:37_1_1_3"/><span class="lab"><font color="Green" title="E14">A17</font></span>: 
 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="c8">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">i9</font>,<font color="Maroon" title="c6">i9</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">i9</font>,<font color="Maroon" title="c10">j9</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_3.html#D9" target="_self" onmouseover="rs('orders_3/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:37_1_1_3"/>
<font color="Maroon" title="c9">f</font> <a href="tarski.html#R2" title="TARSKI: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="c8">j</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c8">j</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">f</font>,<font color="Maroon" title="c7">e</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font></span><br/>

<a NAME="E7:37_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A18</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> (<font color="Maroon" title="c6">i9</font>,<font color="Maroon" title="c10">j9</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_3.html#D9" target="_self" onmouseover="rs('orders_3/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">f9</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c6">i9</font>,<font color="Maroon" title="c10">j9</font><span class="kw"> such that </span><br/><a NAME="E9:37_1_1_3"/><span class="lab"><font color="Green" title="E16">A19</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f9</font>
 <span class="kw">and </span><br/><a NAME="E10:37_1_1_3"/>
<font color="Maroon" title="c11">f9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">i9</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">j9</font>)
 <span class="kw">and </span><br/><a NAME="E11:37_1_1_3"/>
<font color="Maroon" title="c11">f9</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_3.html#D6" target="_self" onmouseover="rs('orders_3/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<a NAME="E12:37_1_1_3"/><span class="lab"><font color="Green" title="E17">A20</font></span>: 
<font color="Maroon" title="c7">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> (<font color="Maroon" title="c6">i9</font>,<font color="Maroon" title="c6">i9</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_3.html#T7" target="_self" onmouseover="rs('orders_3/T7')" onmouseout="rh()">Th7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">e9</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c6">i9</font>,<font color="Maroon" title="c6">i9</font><span class="kw"> such that </span><br/><a NAME="E14:37_1_1_3"/><span class="lab"><font color="Green" title="E18">A21</font></span>: 
<font color="Maroon" title="c7">e</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c12">e9</font>
 <span class="kw">and </span><br/><a NAME="E15:37_1_1_3"/>
<font color="Maroon" title="c12">e9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">i9</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">i9</font>)
 <span class="kw">and </span><br/><a NAME="E16:37_1_1_3"/>
<font color="Maroon" title="c12">e9</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_3.html#D6" target="_self" onmouseover="rs('orders_3/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<a NAME="E17:37_1_1_3"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">f9</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">i9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E18:37_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c11">f9</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">e9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:37_1_1_3"><span class="lab"><font color="Green" title="E18">A21</font></span></a>, <a class="txt" href="#E9:37_1_1_3"><span class="lab"><font color="Green" title="E16">A19</font></span></a>, <a class="ref" href="relat_1.html#T52" onmouseover="rs('relat_1/T52')" onmouseout="rh()">RELAT_1:52</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:37_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="c8">j</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">f</font>,<font color="Maroon" title="c7">e</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:37_1_1_3"><span class="lab"><font color="Green" title="E17">A20</font></span></a>, <a class="txt" href="#E14:37_1_1_3"><span class="lab"><font color="Green" title="E18">A21</font></span></a>, <a class="txt" href="#E7:37_1_1_3"><span class="lab"><font color="Green" title="E15">A18</font></span></a>, <a class="txt" href="#E9:37_1_1_3"><span class="lab"><font color="Green" title="E16">A19</font></span></a>, <a class="txt" href="#E5:37_1_1_3"><span class="lab"><font color="Green" title="E14">A17</font></span></a>, <a class="ref" href="altcat_1.html#T11" onmouseover="rs('altcat_1/T11')" onmouseout="rh()">ALTCAT_1:11</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div>
