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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span>( <font color="Maroon" title="c1">D</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  ex <font color="Olive" title="b1">o</font> being    <a href="flexary1.html#M1" title="FLEXARY1:mode.1">DoubleReorganization</a> of <font color="Maroon" title="c1">D</font> st <font color="Olive" title="b1">o</font> is <font color="Maroon" title="c1">D</font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  );<br/>
<a NAME="E1:60_1_1"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
;<br/>
<a NAME="E2:60_1_1"/><span class="lab"><font color="Green" title="E39">A2</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="60_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">i</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"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">i</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:60_1_1_1"/><span class="lab"><font color="Green" title="E40">A3</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">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"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">assume </span><a NAME="E2:60_1_1_1"/>
<font color="Maroon" title="c3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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" title="b1">o</font> being    <a href="flexary1.html#M1" title="FLEXARY1:mode.1">DoubleReorganization</a> of <font color="Maroon" title="c1">D</font> st <font color="Olive" title="b1">o</font> is <font color="Maroon" title="c3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a> </span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:60_1_1_1_1"/>
( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:60_1_1_1_1_1"/><span class="lab"><font color="Green" title="E41">A4</font></span>: 
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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" title="b1">o</font> being    <a href="flexary1.html#M1" title="FLEXARY1:mode.1">DoubleReorganization</a> of <font color="Maroon" title="c1">D</font> st <font color="Olive" title="b1">o</font> is <font color="Maroon" title="c3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a> </span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c4">F</font> =  <a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <font color="Maroon" title="c1">D</font>;<br/><a NAME="E2:60_1_1_1_1_1"/>
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <font color="Maroon" title="c1">D</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> is    <a href="flexary1.html#M1" title="FLEXARY1:mode.1">DoubleReorganization</a> of <font color="Maroon" title="c1">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="flexary1.html#T35" target="_self" onmouseover="rs('flexary1/T35')" onmouseout="rh()">Th35</a></span>;<br/><span class="kw">hence </span><a NAME="E3:60_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">o</font> being    <a href="flexary1.html#M1" title="FLEXARY1:mode.1">DoubleReorganization</a> of <font color="Maroon" title="c1">D</font> st <font color="Olive" title="b1">o</font> is <font color="Maroon" title="c3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_1_1_1_1_1"><span class="lab"><font color="Green" title="E41">A4</font></span></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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:60_1_1_1_1_2"/>
<font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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" title="b1">o</font> being    <a href="flexary1.html#M1" title="FLEXARY1:mode.1">DoubleReorganization</a> of <font color="Maroon" title="c1">D</font> st <font color="Olive" title="b1">o</font> is <font color="Maroon" title="c3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">o</font> being    <a href="flexary1.html#M1" title="FLEXARY1:mode.1">DoubleReorganization</a> of <font color="Maroon" title="c1">D</font><span class="kw"> such that </span><br/><a NAME="E3:60_1_1_1_1_2"/><span class="lab"><font color="Green" title="E41">A5</font></span>: 
<font color="Maroon" title="c4">o</font> is <font color="Maroon" title="c3">i</font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_1_1_1"><span class="lab"><font color="Green" title="E40">A3</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">e</font> = <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> as    <a href="flexary1.html#M1" title="FLEXARY1:mode.1">DoubleReorganization</a> of  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="flexary1.html#T34" target="_self" onmouseover="rs('flexary1/T34')" onmouseout="rh()">Th34</a></span>;<br/><a NAME="E5:60_1_1_1_1_2"/>
<font color="Maroon" title="c1">D</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E6:60_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">o</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c5">e</font> is    <a href="flexary1.html#M1" title="FLEXARY1:mode.1">DoubleReorganization</a> of <font color="Maroon" title="c1">D</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="flexary1.html#T36" target="_self" onmouseover="rs('flexary1/T36')" onmouseout="rh()">Th36</a></span>;<br/><span class="kw">hence </span><a NAME="E7:60_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">o</font> being    <a href="flexary1.html#M1" title="FLEXARY1:mode.1">DoubleReorganization</a> of <font color="Maroon" title="c1">D</font> st <font color="Olive" title="b1">o</font> is <font color="Maroon" title="c3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60_1_1_1_1_2"><span class="lab"><font color="Green" title="E41">A5</font></span></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><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E3:60_1_1"/>
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E1:60_1_1"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E2:60_1_1"><span class="lab"><font color="Green" title="E39">A2</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:60_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being    <a href="flexary1.html#M1" title="FLEXARY1:mode.1">DoubleReorganization</a> of <font color="Maroon" title="c1">D</font> st <font color="Olive">b<sub>1</sub></font> is <font color="Maroon" title="c2">n</font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</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><br/>


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