<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">X</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span>;<br/>
<a NAME="E1:54"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font> ) &amp; ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> ) )</span><br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:54_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E3:54_1_1"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font>
 <span class="kw">and </span><br/><a NAME="E4:54_1_1"/><span class="lab"><font color="Green" title="E35">A3</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:54_1_1"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> holds <br/><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></span>;<br/><a NAME="E6:54_1_1"/><span class="lab"><font color="Green" title="E37">A5</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54_1_1"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/><a NAME="E7:54_1_1"/><span class="lab"><font color="Green" title="E38">A6</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">g</font> or <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:54_1_1_1"/>
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:54_1_1_1"/><span class="lab"><font color="Green" title="E39">A7</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">g</font>
 <span class="kw">and </span><br/><a NAME="E4:54_1_1_1"/><span class="lab"><font color="Green" title="E40">A8</font></span>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E5:54_1_1_1"/><span class="lab"><font color="Green" title="E41">A9</font></span>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54_1_1"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E5:54_1_1"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="txt" href="#E3:54_1_1_1"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="txt" href="#E4:54_1_1_1"><span class="lab"><font color="Green" title="E40">A8</font></span></a></span>;<br/>
<a NAME="E6:54_1_1_1"/>
 ex <font color="Olive" title="b1">R</font> being   <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a>  st <br/>( <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</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="Olive" title="b1">R</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:54_1_1"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E3:54_1_1_1"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:54_1_1_1"/>
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:54_1_1"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E3:54_1_1_1"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="txt" href="#E5:54_1_1_1"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="ref" href="dickson.html#D15" title="DICKSON:def.15">DICKSON:def 15</a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="lab"><font color="Green" title="E39">A10</font></span>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">g</font> = 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54_1_1"><span class="lab"><font color="Green" title="E35">A3</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>
;<br/><a NAME="E9:54_1_1"/><span class="lab"><font color="Green" title="E40">A11</font></span>: 
<font color="Maroon" title="c4">g</font> is  <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-valued</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:54_1_1"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="ref" href="valued_0.html#D6" title="VALUED_0:def.6">VALUED_0:def 6</a></span>;<br/><a NAME="E10:54_1_1"/>
<font color="Maroon" title="c4">g</font> is   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:54_1_1"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a>, <a class="ref" href="relat_1.html#D18" title="RELAT_1:def.18">RELAT_1:def 18</a></span>;<br/><span class="kw">hence </span><a NAME="E11:54_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_1_1"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E9:54_1_1"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="ref" href="pre_poly.html#D12" title="PRE_POLY:def.12">PRE_POLY:def 12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:54_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">g</font> = <font color="Maroon" title="c3">x</font> as   <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-valued</a>   <a href="pre_poly.html#V2" title="PRE_POLY:attr.2">finite-support</a>  <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D12" title="PRE_POLY:def.12">PRE_POLY:def 12</a></span>;<br/><a NAME="E4:54_1"/><span class="lab"><font color="Green" title="E34">A12</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c5">g</font> = <font color="Maroon" title="c4">g</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> holds <br/><font color="Maroon" title="c5">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) )</span><br/><span class="kw">thus </span><a NAME="E1:54_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> holds <br/><font color="Maroon" title="c5">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) )</span><br/><span class="kw">thus </span><a NAME="E2:54_1_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54_1"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> holds <br/><font color="Maroon" title="c5">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> implies <font color="Maroon" title="c5">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> )</span><br/><span class="kw">assume </span><a NAME="E3:54_1_2"/>
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span><br/><a NAME="E4:54_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A13</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">R</font> being    <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> <span class="kw"> such that </span><br/><a NAME="E6:54_1_2"/><span class="lab"><font color="Green" title="E36">A14</font></span>: 
<font color="Maroon" title="c7">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">and </span><br/><a NAME="E7:54_1_2"/><span class="lab"><font color="Green" title="E37">A15</font></span>: 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</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="c7">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></span>;<br/><a NAME="E8:54_1_2"/>
<font color="Maroon" title="c7">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54_1_2"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="txt" href="#E6:54_1_2"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E9:54_1_2"/>
<font color="Maroon" title="c5">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:54_1_2"><span class="lab"><font color="Green" title="E37">A15</font></span></a>, <a class="ref" href="dickson.html#D15" title="DICKSON:def.15">DICKSON:def 15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E6:54_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E7:54_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:54"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
