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

<span class="kw">let </span><font color="Maroon" title="c1">H1</font>, <font color="Maroon" title="c2">H2</font> be   <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</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">r</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="modelc_2.html#K25" title="MODELC_2:func.25">Inf_seq</a> <a href="modelc_2.html#K43" title="MODELC_2:func.43">AtomicFamily</a> holds <br/> ( <font color="Olive" title="b1">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c2">H2</font> iff <font color="Olive" title="b1">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c2">H2</font> <a href="modelc_2.html#K5" title="MODELC_2:func.5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K6" title="MODELC_2:func.6">'X'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c2">H2</font></span>)</span></span>)</span></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">r</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="modelc_2.html#K25" title="MODELC_2:func.25">Inf_seq</a> <a href="modelc_2.html#K43" title="MODELC_2:func.43">AtomicFamily</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="c3">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c2">H2</font> iff <font color="Maroon" title="c3">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c2">H2</font> <a href="modelc_2.html#K5" title="MODELC_2:func.5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K6" title="MODELC_2:func.6">'X'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c2">H2</font></span>)</span></span>)</span></span>)</span> )</span><br/>





<a NAME="E1:174"/><span class="lab"><font color="Green" title="E123">A1</font></span>: 
( <font color="Maroon" title="c3">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c2">H2</font> <a href="modelc_2.html#K5" title="MODELC_2:func.5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K6" title="MODELC_2:func.6">'X'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c2">H2</font></span>)</span></span>)</span></span>)</span> implies <font color="Maroon" title="c3">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c2">H2</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="174_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:174_1"/><span class="lab"><font color="Green" title="E124">A2</font></span>: 
<font color="Maroon" title="c3">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c2">H2</font> <a href="modelc_2.html#K5" title="MODELC_2:func.5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K6" title="MODELC_2:func.6">'X'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c2">H2</font></span>)</span></span>)</span></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"> <font color="Maroon" title="c3">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c2">H2</font></span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="174_1_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"> <font color="Maroon" title="c3">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c2">H2</font></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="174_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="E1:174_1_1_1"/>
( <font color="Maroon" title="c3">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c2">H2</font> or <font color="Maroon" title="c3">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K6" title="MODELC_2:func.6">'X'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c2">H2</font></span>)</span></span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:174_1"><span class="lab"><font color="Green" title="E124">A2</font></span></a>, <a class="ref" href="modelc_2.html#T66" target="_self" onmouseover="rs('modelc_2/T66')" onmouseout="rh()">Th66</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="174_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:174_1_1_1_1"/><span class="lab"><font color="Green" title="E125">A3</font></span>: 
<font color="Maroon" title="c3">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c2">H2</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="c3">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c2">H2</font></span><br/><div class="add"><a NAME="E2:174_1_1_1_1"/>
 ex <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( (  for <font color="Olive" title="b2">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">r</font>,<font color="Olive" title="b2">j</font>) <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font> ) &amp;  <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">r</font>,<font color="Olive" title="b1">m</font>) <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c2">H2</font> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="174_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
 <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"> ( (  for <font color="Olive" title="b1">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/> <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">r</font>,<font color="Olive" title="b1">j</font>) <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font> ) &amp;  <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">r</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c2">H2</font> )</span><br/>

<span class="kw">thus </span><a NAME="E1:174_1_1_1_1_1"/>
( (  for <font color="Olive" title="b1">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/> <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">r</font>,<font color="Olive" title="b1">j</font>) <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font> ) &amp;  <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">r</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c2">H2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:174_1_1_1_1"><span class="lab"><font color="Green" title="E125">A3</font></span></a>, <a class="txt" href="modelc_2.html#E126"><span class="lab"><font color="Green" title="E82">Lm28</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><span class="kw">hence </span><a NAME="E3:174_1_1_1_1"/>
<font color="Maroon" title="c3">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c2">H2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#T68" target="_self" onmouseover="rs('modelc_2/T68')" onmouseout="rh()">Th68</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="174_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:174_1_1_1_2"/><span class="lab"><font color="Green" title="E125">A4</font></span>: 
<font color="Maroon" title="c3">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K6" title="MODELC_2:func.6">'X'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c2">H2</font></span>)</span></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"> <font color="Maroon" title="c3">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c2">H2</font></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c4">r1</font> =  <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">r</font>,1);<br/><a NAME="E2:174_1_1_1_2"/>
<font color="Maroon" title="c3">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K6" title="MODELC_2:func.6">'X'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c2">H2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:174_1_1_1_2"><span class="lab"><font color="Green" title="E125">A4</font></span></a>, <a class="ref" href="modelc_2.html#T65" target="_self" onmouseover="rs('modelc_2/T65')" onmouseout="rh()">Th65</a></span>;<br/><a NAME="E3:174_1_1_1_2"/><span class="kw">then </span>
 <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">r</font>,1) <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c2">H2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#T67" target="_self" onmouseover="rs('modelc_2/T67')" onmouseout="rh()">Th67</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:174_1_1_1_2"/><span class="lab"><font color="Green" title="E126">A5</font></span>: 
 for <font color="Olive" title="b1">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">m</font> holds <br/> <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">r</font>,1)</span>)</span>,<font color="Olive" title="b1">j</font>) <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font>
 <span class="kw">and </span><br/><a NAME="E6:174_1_1_1_2"/><span class="lab"><font color="Green" title="E127">A6</font></span>: 
 <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">r</font>,1)</span>)</span>,<font color="Maroon" title="c5">m</font>) <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c2">H2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#T68" target="_self" onmouseover="rs('modelc_2/T68')" onmouseout="rh()">Th68</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c6">m1</font> = <font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/><a NAME="E7:174_1_1_1_2"/><span class="lab"><font color="Green" title="E128">A7</font></span>: 
<font color="Maroon" title="c3">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:174_1_1_1_2"><span class="lab"><font color="Green" title="E125">A4</font></span></a>, <a class="ref" href="modelc_2.html#T65" target="_self" onmouseover="rs('modelc_2/T65')" onmouseout="rh()">Th65</a></span>;<br/><a NAME="E8:174_1_1_1_2"/><span class="lab"><font color="Green" title="E129">A8</font></span>: 
 for <font color="Olive" title="b1">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 holds <br/> <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">r</font>,<font color="Olive" title="b1">j</font>) <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="174_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">j</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" title="c7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies  <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c7">j</font>) <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:174_1_1_1_2_1"/><span class="lab"><font color="Green" title="E130">A9</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <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="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c7">j</font>) <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font></span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="174_1_1_1_2_1_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">  <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c7">j</font>) <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="174_1_1_1_2_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="E1:174_1_1_1_2_1_1_1"/>
( <font color="Maroon" title="c7">j</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="c7">j</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="174_1_1_1_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:174_1_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c7">j</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">  <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c7">j</font>) <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:174_1_1_1_2_1_1_1_1"/>
 <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c7">j</font>) <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:174_1_1_1_2"><span class="lab"><font color="Green" title="E128">A7</font></span></a>, <a class="txt" href="modelc_2.html#E126"><span class="lab"><font color="Green" title="E82">Lm28</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="174_1_1_1_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:174_1_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E131">A10</font></span>: 
<font color="Maroon" title="c7">j</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">  <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c7">j</font>) <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c8">j1</font> = <font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">j1</font> = <font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:174_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E131">A10</font></span></a>, <a class="ref" href="nat_1.html#T20" onmouseover="rs('nat_1/T20')" onmouseout="rh()">NAT_1:20</a></span>;<br/><a NAME="E3:174_1_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:174_1_1_1_2_1"><span class="lab"><font color="Green" title="E130">A9</font></span></a>, <a class="ref" href="xreal_1.html#T14" onmouseover="rs('xreal_1/T14')" onmouseout="rh()">XREAL_1:14</a></span>;<br/><a NAME="E4:174_1_1_1_2_1_1_1_2"/><span class="kw">then </span>
 <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">r</font>,1)</span>)</span>,<font color="Maroon" title="c9">j1</font>) <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:174_1_1_1_2"><span class="lab"><font color="Green" title="E126">A5</font></span></a></span>;<br/><a NAME="E5:174_1_1_1_2_1_1_1_2"/><span class="kw">then </span>
 <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">r</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="modelc_2.html#E127"><span class="lab"><font color="Green" title="E83">Lm29</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:174_1_1_1_2_1_1_1_2"/>
 <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c7">j</font>) <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</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"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:174_1_1_1_2_1"/>
 <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c7">j</font>) <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</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"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E9:174_1_1_1_2"/>
 <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">r</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c2">H2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:174_1_1_1_2"><span class="lab"><font color="Green" title="E127">A6</font></span></a>, <a class="txt" href="modelc_2.html#E127"><span class="lab"><font color="Green" title="E83">Lm29</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E10:174_1_1_1_2"/>
<font color="Maroon" title="c3">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c2">H2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:174_1_1_1_2"><span class="lab"><font color="Green" title="E129">A8</font></span></a>, <a class="ref" href="modelc_2.html#T68" target="_self" onmouseover="rs('modelc_2/T68')" onmouseout="rh()">Th68</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>
<span class="kw">hence </span><a NAME="E3:174_1"/>
<font color="Maroon" title="c3">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c2">H2</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:174"/>
( <font color="Maroon" title="c3">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c2">H2</font> implies <font color="Maroon" title="c3">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c2">H2</font> <a href="modelc_2.html#K5" title="MODELC_2:func.5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K6" title="MODELC_2:func.6">'X'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c2">H2</font></span>)</span></span>)</span></span>)</span> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="174_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:174_2"/>
<font color="Maroon" title="c3">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c2">H2</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="c3">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c2">H2</font> <a href="modelc_2.html#K5" title="MODELC_2:func.5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K6" title="MODELC_2:func.6">'X'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c2">H2</font></span>)</span></span>)</span></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:174_2"/><span class="lab"><font color="Green" title="E124">A11</font></span>: 
 for <font color="Olive" title="b1">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">m</font> holds <br/> <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">r</font>,<font color="Olive" title="b1">j</font>) <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font>
 <span class="kw">and </span><br/><a NAME="E4:174_2"/><span class="lab"><font color="Green" title="E125">A12</font></span>: 
 <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">m</font>) <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c2">H2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#T68" target="_self" onmouseover="rs('modelc_2/T68')" onmouseout="rh()">Th68</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="174_2_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="E1:174_2_1"/>
( <font color="Maroon" title="c4">m</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="c4">m</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="174_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:174_2_1_1"/>
<font color="Maroon" title="c4">m</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"> <font color="Maroon" title="c3">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c2">H2</font> <a href="modelc_2.html#K5" title="MODELC_2:func.5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K6" title="MODELC_2:func.6">'X'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c2">H2</font></span>)</span></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:174_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c2">H2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:174_2"><span class="lab"><font color="Green" title="E125">A12</font></span></a>, <a class="txt" href="modelc_2.html#E126"><span class="lab"><font color="Green" title="E82">Lm28</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:174_2_1_1"/>
<font color="Maroon" title="c3">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c2">H2</font> <a href="modelc_2.html#K5" title="MODELC_2:func.5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K6" title="MODELC_2:func.6">'X'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c2">H2</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#T66" target="_self" onmouseover="rs('modelc_2/T66')" onmouseout="rh()">Th66</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="174_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:174_2_1_2"/><span class="lab"><font color="Green" title="E126">A13</font></span>: 
<font color="Maroon" title="c4">m</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"> <font color="Maroon" title="c3">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c2">H2</font> <a href="modelc_2.html#K5" title="MODELC_2:func.5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K6" title="MODELC_2:func.6">'X'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c2">H2</font></span>)</span></span>)</span></span>)</span></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c5">k</font> = <font color="Maroon" title="c4">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">k</font> = <font color="Maroon" title="c4">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:174_2_1_2"><span class="lab"><font color="Green" title="E126">A13</font></span></a>, <a class="ref" href="nat_1.html#T20" onmouseover="rs('nat_1/T20')" onmouseout="rh()">NAT_1:20</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c7">r1</font> =  <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">r</font>,1);<br/><a NAME="E3:174_2_1_2"/><span class="lab"><font color="Green" title="E127">A14</font></span>: 
 for <font color="Olive" title="b1">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> holds <br/> <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">r</font>,1)</span>)</span>,<font color="Olive" title="b1">j</font>) <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="174_2_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">j</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" title="c8">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> implies  <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">r</font>,1)</span>)</span>,<font color="Maroon" title="c8">j</font>) <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:174_2_1_2_1"/>
<font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</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">  <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">r</font>,1)</span>)</span>,<font color="Maroon" title="c8">j</font>) <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font></span><br/>

<a NAME="E2:174_2_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E128">A15</font></span>: 
<font color="Maroon" title="c8">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/>
<a NAME="E3:174_2_1_2_1"/>
 <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">r</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">r</font>,1)</span>)</span>,<font color="Maroon" title="c8">j</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="modelc_2.html#E127"><span class="lab"><font color="Green" title="E83">Lm29</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:174_2_1_2_1"/>
 <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">r</font>,1)</span>)</span>,<font color="Maroon" title="c8">j</font>) <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:174_2"><span class="lab"><font color="Green" title="E124">A11</font></span></a>, <a class="txt" href="#E2:174_2_1_2_1"><span class="lab"><font color="Green" title="E128">A15</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><a NAME="E4:174_2_1_2"/>
 <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">r</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">r</font>,1)</span>)</span>,<font color="Maroon" title="c6">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="modelc_2.html#E127"><span class="lab"><font color="Green" title="E83">Lm29</font></span></a></span>;<br/><a NAME="E5:174_2_1_2"/><span class="kw">then </span>
 <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">r</font>,1) <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c2">H2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:174_2"><span class="lab"><font color="Green" title="E125">A12</font></span></a>, <a class="txt" href="#E3:174_2_1_2"><span class="lab"><font color="Green" title="E127">A14</font></span></a>, <a class="ref" href="modelc_2.html#T68" target="_self" onmouseover="rs('modelc_2/T68')" onmouseout="rh()">Th68</a></span>;<br/><a NAME="E6:174_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E128">A16</font></span>: 
<font color="Maroon" title="c3">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K6" title="MODELC_2:func.6">'X'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c2">H2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#T67" target="_self" onmouseover="rs('modelc_2/T67')" onmouseout="rh()">Th67</a></span>;<br/><a NAME="E7:174_2_1_2"/>
 <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">r</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="modelc_2.html#E126"><span class="lab"><font color="Green" title="E82">Lm28</font></span></a></span>;<br/><a NAME="E8:174_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:174_2"><span class="lab"><font color="Green" title="E124">A11</font></span></a>, <a class="txt" href="#E1:174_2_1_2"><span class="lab"><font color="Green" title="E126">A13</font></span></a></span>;<br/><a NAME="E9:174_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K6" title="MODELC_2:func.6">'X'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c2">H2</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:174_2_1_2"><span class="lab"><font color="Green" title="E128">A16</font></span></a>, <a class="ref" href="modelc_2.html#T65" target="_self" onmouseover="rs('modelc_2/T65')" onmouseout="rh()">Th65</a></span>;<br/><span class="kw">hence </span><a NAME="E10:174_2_1_2"/>
<font color="Maroon" title="c3">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c2">H2</font> <a href="modelc_2.html#K5" title="MODELC_2:func.5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K6" title="MODELC_2:func.6">'X'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c2">H2</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#T66" target="_self" onmouseover="rs('modelc_2/T66')" onmouseout="rh()">Th66</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>
<span class="kw">hence </span><a NAME="E3:174"/>
( <font color="Maroon" title="c3">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c2">H2</font> iff <font color="Maroon" title="c3">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c2">H2</font> <a href="modelc_2.html#K5" title="MODELC_2:func.5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K6" title="MODELC_2:func.6">'X'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c2">H2</font></span>)</span></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:174"><span class="lab"><font color="Green" title="E123">A1</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>
