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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">S</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">BASSIGN</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K30" title="MODELC_1:func.30">ModelSP</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K25" title="MODELC_2:func.25">Inf_seq</a> <font color="Maroon" title="c1">S</font></span>)</span></span>)</span><br/>  for <font color="Olive" title="b2">t</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> <font color="Maroon" title="c1">S</font><br/>  for <font color="Olive" title="b3">f</font>, <font color="Olive" title="b4">g</font> being   <a href="modelc_2.html#NM2" title="MODELC_2:NM.2">Assign</a> of <span class="p1">(<span class="default"><a href="modelc_2.html#K42" title="MODELC_2:func.42">Inf_seqModel</a> (<font color="Maroon" title="c1">S</font>,<font color="Olive" title="b1">BASSIGN</font>)</span>)</span> holds <br/> ( <font color="Olive" title="b2">t</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Olive" title="b3">f</font> <a href="modelc_2.html#K23" title="MODELC_2:func.23">'U'</a> <font color="Olive" title="b4">g</font> iff  ex <font color="Olive" title="b5">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( (  for <font color="Olive" title="b6">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b6">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">m</font> holds <br/> <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Olive" title="b2">t</font>,<font color="Olive" title="b6">j</font>) <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Olive" title="b3">f</font> ) &amp;  <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Olive" title="b2">t</font>,<font color="Olive" title="b5">m</font>) <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Olive" title="b4">g</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">BASSIGN</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K30" title="MODELC_1:func.30">ModelSP</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K25" title="MODELC_2:func.25">Inf_seq</a> <font color="Maroon" title="c1">S</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">  for <font color="Olive" title="b1">t</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> <font color="Maroon" title="c1">S</font><br/>  for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</font> being   <a href="modelc_2.html#NM2" title="MODELC_2:NM.2">Assign</a> of <span class="p1">(<span class="default"><a href="modelc_2.html#K42" title="MODELC_2:func.42">Inf_seqModel</a> (<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">BASSIGN</font>)</span>)</span> holds <br/> ( <font color="Olive" title="b1">t</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Olive" title="b2">f</font> <a href="modelc_2.html#K23" title="MODELC_2:func.23">'U'</a> <font color="Olive" title="b3">g</font> iff  ex <font color="Olive" title="b4">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( (  for <font color="Olive" title="b5">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b5">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">m</font> holds <br/> <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Olive" title="b1">t</font>,<font color="Olive" title="b5">j</font>) <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Olive" title="b2">f</font> ) &amp;  <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Olive" title="b1">t</font>,<font color="Olive" title="b4">m</font>) <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Olive" title="b3">g</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">t</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> <font color="Maroon" title="c1">S</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">  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="modelc_2.html#NM2" title="MODELC_2:NM.2">Assign</a> of <span class="p1">(<span class="default"><a href="modelc_2.html#K42" title="MODELC_2:func.42">Inf_seqModel</a> (<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">BASSIGN</font>)</span>)</span> holds <br/> ( <font color="Maroon" title="c3">t</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Olive" title="b1">f</font> <a href="modelc_2.html#K23" title="MODELC_2:func.23">'U'</a> <font color="Olive" title="b2">g</font> iff  ex <font color="Olive" title="b3">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( (  for <font color="Olive" title="b4">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b4">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">m</font> holds <br/> <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">t</font>,<font color="Olive" title="b4">j</font>) <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Olive" title="b1">f</font> ) &amp;  <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">t</font>,<font color="Olive" title="b3">m</font>) <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Olive" title="b2">g</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</font> be   <a href="modelc_2.html#NM2" title="MODELC_2:NM.2">Assign</a> of <span class="p1">(<span class="default"><a href="modelc_2.html#K42" title="MODELC_2:func.42">Inf_seqModel</a> (<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">BASSIGN</font>)</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">t</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c4">f</font> <a href="modelc_2.html#K23" title="MODELC_2:func.23">'U'</a> <font color="Maroon" title="c5">g</font> iff  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">t</font>,<font color="Olive" title="b2">j</font>) <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c4">f</font> ) &amp;  <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">t</font>,<font color="Olive" title="b1">m</font>) <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c5">g</font> ) )</span><br/>









<span class="kw">set </span><font color="Maroon" title="c6">S1</font> =  <a href="modelc_2.html#K25" title="MODELC_2:func.25">Inf_seq</a> <font color="Maroon" title="c1">S</font>;<br/>
<a NAME="E1:149"/><span class="lab"><font color="Green" title="E106">A1</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="modelc_2.html#K23" title="MODELC_2:func.23">'U'</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_2.html#K38" title="MODELC_2:func.38">Until_0</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c1">S</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#D53" target="_self" onmouseover="rs('modelc_2/D53')" onmouseout="rh()">Def53</a></span>;<br/>
<a NAME="E2:149"/><span class="lab"><font color="Green" title="E107">A2</font></span>: 
(  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">t</font>,<font color="Olive" title="b2">j</font>) <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c4">f</font> ) &amp;  <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">t</font>,<font color="Olive" title="b1">m</font>) <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c5">g</font> ) implies <font color="Maroon" title="c3">t</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c4">f</font> <a href="modelc_2.html#K23" title="MODELC_2:func.23">'U'</a> <font color="Maroon" title="c5">g</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="149_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:149_1"/><span class="lab"><font color="Green" title="E108">A3</font></span>: 
 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">t</font>,<font color="Olive" title="b2">j</font>) <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c4">f</font> ) &amp;  <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">t</font>,<font color="Olive" title="b1">m</font>) <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c5">g</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">t</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c4">f</font> <a href="modelc_2.html#K23" title="MODELC_2:func.23">'U'</a> <font color="Maroon" title="c5">g</font></span><br/>

<a NAME="E2:149_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/><span class="p1">(<span class="default"><a href="modelc_1.html#K31" title="MODELC_1:func.31">Fid</a> (<font color="Maroon" title="c4">f</font>,<span class="p2">(<span class="default"><a href="modelc_2.html#K25" title="MODELC_2:func.25">Inf_seq</a> <font color="Maroon" title="c1">S</font></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K28" title="MODELC_2:func.28">Shift</a> (<font color="Maroon" title="c3">t</font>,<font color="Olive" title="b2">j</font>,<font color="Maroon" title="c1">S</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a>  ) &amp; <span class="p1">(<span class="default"><a href="modelc_1.html#K31" title="MODELC_1:func.31">Fid</a> (<font color="Maroon" title="c5">g</font>,<span class="p2">(<span class="default"><a href="modelc_2.html#K25" title="MODELC_2:func.25">Inf_seq</a> <font color="Maroon" title="c1">S</font></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K28" title="MODELC_2:func.28">Shift</a> (<font color="Maroon" title="c3">t</font>,<font color="Olive" title="b1">m</font>,<font color="Maroon" title="c1">S</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="149_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c7">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E2:149_1_1"/><span class="lab"><font color="Green" title="E109">A4</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="c7">m</font> holds <br/> <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">t</font>,<font color="Olive" title="b1">j</font>) <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">and </span><br/><a NAME="E3:149_1_1"/><span class="lab"><font color="Green" title="E110">A5</font></span>: 
 <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">m</font>) <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c5">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:149_1"><span class="lab"><font color="Green" title="E108">A3</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">m</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"> ( (  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="c7">m</font> holds <br/><span class="p1">(<span class="default"><a href="modelc_1.html#K31" title="MODELC_1:func.31">Fid</a> (<font color="Maroon" title="c4">f</font>,<span class="p2">(<span class="default"><a href="modelc_2.html#K25" title="MODELC_2:func.25">Inf_seq</a> <font color="Maroon" title="c1">S</font></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K28" title="MODELC_2:func.28">Shift</a> (<font color="Maroon" title="c3">t</font>,<font color="Olive" title="b1">j</font>,<font color="Maroon" title="c1">S</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a>  ) &amp; <span class="p1">(<span class="default"><a href="modelc_1.html#K31" title="MODELC_1:func.31">Fid</a> (<font color="Maroon" title="c5">g</font>,<span class="p2">(<span class="default"><a href="modelc_2.html#K25" title="MODELC_2:func.25">Inf_seq</a> <font color="Maroon" title="c1">S</font></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K28" title="MODELC_2:func.28">Shift</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">m</font>,<font color="Maroon" title="c1">S</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a>  )</span><br/>

<a NAME="E4:149_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> <font color="Maroon" title="c7">m</font> holds <br/><span class="p1">(<span class="default"><a href="modelc_1.html#K31" title="MODELC_1:func.31">Fid</a> (<font color="Maroon" title="c4">f</font>,<span class="p2">(<span class="default"><a href="modelc_2.html#K25" title="MODELC_2:func.25">Inf_seq</a> <font color="Maroon" title="c1">S</font></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K28" title="MODELC_2:func.28">Shift</a> (<font color="Maroon" title="c3">t</font>,<font color="Olive" title="b1">j</font>,<font color="Maroon" title="c1">S</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="149_1_1_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="c7">m</font> implies <span class="p1">(<span class="default"><a href="modelc_1.html#K31" title="MODELC_1:func.31">Fid</a> (<font color="Maroon" title="c4">f</font>,<span class="p2">(<span class="default"><a href="modelc_2.html#K25" title="MODELC_2:func.25">Inf_seq</a> <font color="Maroon" title="c1">S</font></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K28" title="MODELC_2:func.28">Shift</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c1">S</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:149_1_1_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="c7">m</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"><a href="modelc_1.html#K31" title="MODELC_1:func.31">Fid</a> (<font color="Maroon" title="c4">f</font>,<span class="p2">(<span class="default"><a href="modelc_2.html#K25" title="MODELC_2:func.25">Inf_seq</a> <font color="Maroon" title="c1">S</font></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K28" title="MODELC_2:func.28">Shift</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c1">S</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a> </span><br/>

<a NAME="E2:149_1_1_1"/><span class="kw">then </span>
 <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c8">j</font>) <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:149_1_1"><span class="lab"><font color="Green" title="E109">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:149_1_1_1"/>
<span class="p1">(<span class="default"><a href="modelc_1.html#K31" title="MODELC_1:func.31">Fid</a> (<font color="Maroon" title="c4">f</font>,<span class="p2">(<span class="default"><a href="modelc_2.html#K25" title="MODELC_2:func.25">Inf_seq</a> <font color="Maroon" title="c1">S</font></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K28" title="MODELC_2:func.28">Shift</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c1">S</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</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/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:149_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> <font color="Maroon" title="c7">m</font> holds <br/><span class="p1">(<span class="default"><a href="modelc_1.html#K31" title="MODELC_1:func.31">Fid</a> (<font color="Maroon" title="c4">f</font>,<span class="p2">(<span class="default"><a href="modelc_2.html#K25" title="MODELC_2:func.25">Inf_seq</a> <font color="Maroon" title="c1">S</font></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K28" title="MODELC_2:func.28">Shift</a> (<font color="Maroon" title="c3">t</font>,<font color="Olive" title="b1">j</font>,<font color="Maroon" title="c1">S</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a>  ) &amp; <span class="p1">(<span class="default"><a href="modelc_1.html#K31" title="MODELC_1:func.31">Fid</a> (<font color="Maroon" title="c5">g</font>,<span class="p2">(<span class="default"><a href="modelc_2.html#K25" title="MODELC_2:func.25">Inf_seq</a> <font color="Maroon" title="c1">S</font></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K28" title="MODELC_2:func.28">Shift</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">m</font>,<font color="Maroon" title="c1">S</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:149_1_1"><span class="lab"><font color="Green" title="E110">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>
<a NAME="E3:149_1"/><span class="kw">then </span>
 <a href="modelc_2.html#K37" title="MODELC_2:func.37">Until_univ</a> (<font color="Maroon" title="c3">t</font>,<span class="p1">(<span class="default"><a href="modelc_1.html#K31" title="MODELC_1:func.31">Fid</a> (<font color="Maroon" title="c4">f</font>,<span class="p2">(<span class="default"><a href="modelc_2.html#K25" title="MODELC_2:func.25">Inf_seq</a> <font color="Maroon" title="c1">S</font></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="modelc_1.html#K31" title="MODELC_1:func.31">Fid</a> (<font color="Maroon" title="c5">g</font>,<span class="p2">(<span class="default"><a href="modelc_2.html#K25" title="MODELC_2:func.25">Inf_seq</a> <font color="Maroon" title="c1">S</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c1">S</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#D51" target="_self" onmouseover="rs('modelc_2/D51')" onmouseout="rh()">Def51</a></span>;<br/>
<a NAME="E4:149_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="modelc_1.html#K31" title="MODELC_1:func.31">Fid</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="modelc_2.html#K23" title="MODELC_2:func.23">'U'</a> <font color="Maroon" title="c5">g</font></span>)</span>,<span class="p2">(<span class="default"><a href="modelc_2.html#K25" title="MODELC_2:func.25">Inf_seq</a> <font color="Maroon" title="c1">S</font></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:149"><span class="lab"><font color="Green" title="E106">A1</font></span></a>, <a class="ref" href="modelc_2.html#D52" target="_self" onmouseover="rs('modelc_2/D52')" onmouseout="rh()">Def52</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:149_1"/>
<font color="Maroon" title="c3">t</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c4">f</font> <a href="modelc_2.html#K23" title="MODELC_2:func.23">'U'</a> <font color="Maroon" title="c5">g</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="E3:149"/>
( <font color="Maroon" title="c3">t</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c4">f</font> <a href="modelc_2.html#K23" title="MODELC_2:func.23">'U'</a> <font color="Maroon" title="c5">g</font> implies  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">t</font>,<font color="Olive" title="b2">j</font>) <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c4">f</font> ) &amp;  <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">t</font>,<font color="Olive" title="b1">m</font>) <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c5">g</font> ) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="149_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:149_2"/>
<font color="Maroon" title="c3">t</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c4">f</font> <a href="modelc_2.html#K23" title="MODELC_2:func.23">'U'</a> <font color="Maroon" title="c5">g</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">  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">t</font>,<font color="Olive" title="b2">j</font>) <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c4">f</font> ) &amp;  <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">t</font>,<font color="Olive" title="b1">m</font>) <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c5">g</font> )</span><br/>

<a NAME="E2:149_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="modelc_1.html#K31" title="MODELC_1:func.31">Fid</a> (<span class="p2">(<span class="default"><a href="modelc_2.html#K38" title="MODELC_2:func.38">Until_0</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c1">S</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="modelc_2.html#K25" title="MODELC_2:func.25">Inf_seq</a> <font color="Maroon" title="c1">S</font></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:149"><span class="lab"><font color="Green" title="E106">A1</font></span></a></span>;<br/>
<a NAME="E3:149_2"/><span class="kw">then </span>
 <a href="modelc_2.html#K37" title="MODELC_2:func.37">Until_univ</a> (<font color="Maroon" title="c3">t</font>,<span class="p1">(<span class="default"><a href="modelc_1.html#K31" title="MODELC_1:func.31">Fid</a> (<font color="Maroon" title="c4">f</font>,<span class="p2">(<span class="default"><a href="modelc_2.html#K25" title="MODELC_2:func.25">Inf_seq</a> <font color="Maroon" title="c1">S</font></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="modelc_1.html#K31" title="MODELC_1:func.31">Fid</a> (<font color="Maroon" title="c5">g</font>,<span class="p2">(<span class="default"><a href="modelc_2.html#K25" title="MODELC_2:func.25">Inf_seq</a> <font color="Maroon" title="c1">S</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c1">S</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#D52" target="_self" onmouseover="rs('modelc_2/D52')" onmouseout="rh()">Def52</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:149_2"/><span class="lab"><font color="Green" title="E108">A6</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="c7">m</font> holds <br/><span class="p1">(<span class="default"><a href="modelc_1.html#K31" title="MODELC_1:func.31">Fid</a> (<font color="Maroon" title="c4">f</font>,<span class="p2">(<span class="default"><a href="modelc_2.html#K25" title="MODELC_2:func.25">Inf_seq</a> <font color="Maroon" title="c1">S</font></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K28" title="MODELC_2:func.28">Shift</a> (<font color="Maroon" title="c3">t</font>,<font color="Olive" title="b1">j</font>,<font color="Maroon" title="c1">S</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a> 
 <span class="kw">and </span><br/><a NAME="E6:149_2"/><span class="lab"><font color="Green" title="E109">A7</font></span>: 
<span class="p1">(<span class="default"><a href="modelc_1.html#K31" title="MODELC_1:func.31">Fid</a> (<font color="Maroon" title="c5">g</font>,<span class="p2">(<span class="default"><a href="modelc_2.html#K25" title="MODELC_2:func.25">Inf_seq</a> <font color="Maroon" title="c1">S</font></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K28" title="MODELC_2:func.28">Shift</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">m</font>,<font color="Maroon" title="c1">S</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#D51" target="_self" onmouseover="rs('modelc_2/D51')" onmouseout="rh()">Def51</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">m</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"> ( (  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="c7">m</font> holds <br/> <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">t</font>,<font color="Olive" title="b1">j</font>) <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c4">f</font> ) &amp;  <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">m</font>) <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c5">g</font> )</span><br/>

<a NAME="E7:149_2"/>
 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="c7">m</font> holds <br/> <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">t</font>,<font color="Olive" title="b1">j</font>) <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:149_2"><span class="lab"><font color="Green" title="E108">A6</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:149_2"/>
( (  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="c7">m</font> holds <br/> <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">t</font>,<font color="Olive" title="b1">j</font>) <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c4">f</font> ) &amp;  <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">m</font>) <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c5">g</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:149_2"><span class="lab"><font color="Green" title="E109">A7</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="E4:149"/>
( <font color="Maroon" title="c3">t</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c4">f</font> <a href="modelc_2.html#K23" title="MODELC_2:func.23">'U'</a> <font color="Maroon" title="c5">g</font> iff  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">t</font>,<font color="Olive" title="b2">j</font>) <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c4">f</font> ) &amp;  <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c3">t</font>,<font color="Olive" title="b1">m</font>) <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c5">g</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:149"><span class="lab"><font color="Green" title="E107">A2</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>
