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<span class="kw">let </span><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">j</font>, <font color="Maroon" title="c3">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c3">n</font> holds  (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>) <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>) <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>) <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Olive" title="b2">b</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> be   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c3">n</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="c1">i</font>,<font color="Maroon" title="c2">j</font>) <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>) <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>) <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c5">b</font></span>)</span></span><br/>



<span class="kw">set </span><font color="Maroon" title="c6">CUTAB</font> = (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>) <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c5">b</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">CUTA</font> = (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>) <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c4">a</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">CUTB</font> = (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>) <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c5">b</font>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="25_1"><span class="kw">now </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" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">i</font> holds <br/><span class="p1">(<span class="default">(<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>) <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c5">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>) <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>) <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c5">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">i</font> implies <span class="p1">(<span class="default">(<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>) <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c5">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>) <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>) <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c5">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:25_1"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">i</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">(<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>) <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c5">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>) <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>) <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c5">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span><br/><a NAME="E2:25_1"/>
<font color="Maroon" title="c2">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">i</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="axioms.html#T4" onmouseover="rs('axioms/T4')" onmouseout="rh()">AXIOMS:4</a></span>;<br/><a NAME="E3:25_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_1"><span class="lab"><font color="Green" title="E16">A1</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">x9</font> = <font color="Maroon" title="c9">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/><a NAME="E5:25_1"/>
<span class="p1">(<span class="default">(<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>) <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c5">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">x9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_1"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="bagorder.html#D1" target="_self" onmouseover="rs('bagorder/D1')" onmouseout="rh()">Def1</a></span>;<br/><a NAME="E6:25_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A2</font></span>: 
<span class="p1">(<span class="default">(<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>) <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c5">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">x9</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">x9</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D5" onmouseover="rs('pre_poly/D5')" onmouseout="rh()">PRE_POLY:def 5</a></span>;<br/><a NAME="E7:25_1"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
<span class="p1">(<span class="default">(<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>) <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">x9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_1"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="bagorder.html#D1" target="_self" onmouseover="rs('bagorder/D1')" onmouseout="rh()">Def1</a></span>;<br/><a NAME="E8:25_1"/>
<span class="p1">(<span class="default">(<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>) <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">x9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_1"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="bagorder.html#D1" target="_self" onmouseover="rs('bagorder/D1')" onmouseout="rh()">Def1</a></span>;<br/><span class="kw">hence </span><a NAME="E9:25_1"/>
<span class="p1">(<span class="default">(<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>) <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c5">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>) <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>) <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c5">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E7:25_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="pre_poly.html#D5" onmouseover="rs('pre_poly/D5')" onmouseout="rh()">PRE_POLY:def 5</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">hence </span><a NAME="E2:25"/>
(<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>) <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>) <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>) <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c5">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T3" onmouseover="rs('pboole/T3')" onmouseout="rh()">PBOOLE:3</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/>


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