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<span class="kw">let </span><font color="Maroon" title="c1">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/>  for <font color="Olive" title="b2">S</font> being   <a href="fomodel1.html#NM2" title="FOMODEL1:NM.2">Language</a><br/>  for <font color="Olive" title="b3">U</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/>  for <font color="Olive" title="b4">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">(<font color="Olive" title="b2">S</font>,<font color="Olive" title="b3">U</font>) <a href="fomodel2.html#K23" title="FOMODEL2:func.23">-TruthEval</a> <font color="Maroon" title="c1">m</font></span>)</span> holds <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">(<font color="Olive" title="b2">S</font>,<font color="Olive" title="b3">U</font>) <a href="fomodel2.html#K23" title="FOMODEL2:func.23">-TruthEval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b4">n</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default">(<font color="Olive" title="b2">S</font>,<font color="Olive" title="b3">U</font>) <a href="fomodel2.html#K23" title="FOMODEL2:func.23">-TruthEval</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Olive" title="b2">S</font>,<font color="Olive" title="b3">U</font>) <a href="fomodel2.html#K23" title="FOMODEL2:func.23">-TruthEval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b4">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   <a href="fomodel1.html#NM2" title="FOMODEL1:NM.2">Language</a><br/>  for <font color="Olive" title="b2">U</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/>  for <font color="Olive" title="b3">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">(<font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">U</font>) <a href="fomodel2.html#K23" title="FOMODEL2:func.23">-TruthEval</a> <font color="Maroon" title="c1">m</font></span>)</span> holds <br/>( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">(<font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">U</font>) <a href="fomodel2.html#K23" title="FOMODEL2:func.23">-TruthEval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b3">n</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default">(<font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">U</font>) <a href="fomodel2.html#K23" title="FOMODEL2:func.23">-TruthEval</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">U</font>) <a href="fomodel2.html#K23" title="FOMODEL2:func.23">-TruthEval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b3">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">S</font> be   <a href="fomodel1.html#NM2" title="FOMODEL1:NM.2">Language</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">U</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/>  for <font color="Olive" title="b2">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Olive" title="b1">U</font>) <a href="fomodel2.html#K23" title="FOMODEL2:func.23">-TruthEval</a> <font color="Maroon" title="c1">m</font></span>)</span> holds <br/>( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Olive" title="b1">U</font>) <a href="fomodel2.html#K23" title="FOMODEL2:func.23">-TruthEval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">n</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Olive" title="b1">U</font>) <a href="fomodel2.html#K23" title="FOMODEL2:func.23">-TruthEval</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Olive" title="b1">U</font>) <a href="fomodel2.html#K23" title="FOMODEL2:func.23">-TruthEval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">U</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K23" title="FOMODEL2:func.23">-TruthEval</a> <font color="Maroon" title="c1">m</font></span>)</span> holds <br/>( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K23" title="FOMODEL2:func.23">-TruthEval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K23" title="FOMODEL2:func.23">-TruthEval</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K23" title="FOMODEL2:func.23">-TruthEval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">f</font> = (<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K22" title="FOMODEL2:func.22">-TruthEval</a> ;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K23" title="FOMODEL2:func.23">-TruthEval</a> <font color="Maroon" title="c1">m</font></span>)</span> implies ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K23" title="FOMODEL2:func.23">-TruthEval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> $1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K23" title="FOMODEL2:func.23">-TruthEval</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K23" title="FOMODEL2:func.23">-TruthEval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> $1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> ) );<br/>
<a NAME="E1:59"/><span class="lab"><font color="Green" title="E25">B0</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E2:59"/><span class="lab"><font color="Green" title="E26">B1</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">k</font> = <font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">n</font>, <font color="Maroon" title="c8">K</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:59_1"/><span class="lab"><font color="Green" title="E27">C1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font>]
 ; <a class="txt" 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="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">set </span><font color="Maroon" title="c9">g</font> = (<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K23" title="FOMODEL2:func.23">-TruthEval</a> <font color="Maroon" title="c1">m</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">g0</font> = (<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K23" title="FOMODEL2:func.23">-TruthEval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">n</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">g1</font> = (<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K23" title="FOMODEL2:func.23">-TruthEval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/>
<a NAME="E3:59_1"/><span class="lab"><font color="Green" title="E28">C2</font></span>: 
( <span class="p1">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K22" title="FOMODEL2:func.22">-TruthEval</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K23" title="FOMODEL2:func.23">-TruthEval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">n</font></span>)</span> &amp; <span class="p1">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K22" title="FOMODEL2:func.22">-TruthEval</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K23" title="FOMODEL2:func.23">-TruthEval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel2.html#D21" target="_self" title="FOMODEL2:def.21">DefTruth2</a></span>;<br/>
<a NAME="E4:59_1"/><span class="lab"><font color="Green" title="E29">C6</font></span>: 
<span class="p1">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K22" title="FOMODEL2:func.22">-TruthEval</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel2.html#K19" title="FOMODEL2:func.19">ExIterator</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K22" title="FOMODEL2:func.22">-TruthEval</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">k</font></span>)</span></span>)</span> <a href="funct_4.html#K7" title="FUNCT_4:func.7">+*</a> <span class="p2">(<span class="default"><a href="fomodel2.html#K21" title="FOMODEL2:func.21">NorIterator</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K22" title="FOMODEL2:func.22">-TruthEval</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K22" title="FOMODEL2:func.22">-TruthEval</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel2.html#D20" target="_self" title="FOMODEL2:def.20">DefTruth1</a></span>;<br/>
<a NAME="E5:59_1"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K22" title="FOMODEL2:func.22">-TruthEval</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fomodel2.html#K19" title="FOMODEL2:func.19">ExIterator</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K22" title="FOMODEL2:func.22">-TruthEval</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">k</font></span>)</span></span>)</span> <a href="funct_4.html#K7" title="FUNCT_4:func.7">+*</a> <span class="p3">(<span class="default"><a href="fomodel2.html#K21" title="FOMODEL2:func.21">NorIterator</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K22" title="FOMODEL2:func.22">-TruthEval</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K22" title="FOMODEL2:func.22">-TruthEval</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">k</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>;<br/>
<a NAME="E6:59_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">C3</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K22" title="FOMODEL2:func.22">-TruthEval</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K22" title="FOMODEL2:func.22">-TruthEval</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">K</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<span class="kw">assume </span><a NAME="E7:59_1"/><span class="lab"><font color="Green" title="E31">C8</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K23" title="FOMODEL2:func.23">-TruthEval</a> <font color="Maroon" title="c1">m</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K23" title="FOMODEL2:func.23">-TruthEval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K23" title="FOMODEL2:func.23">-TruthEval</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K23" title="FOMODEL2:func.23">-TruthEval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> )</span><br/>

<a NAME="E8:59_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K22" title="FOMODEL2:func.22">-TruthEval</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1"><span class="lab"><font color="Green" title="E27">C1</font></span></a>, <a class="ref" href="fomodel2.html#D21" target="_self" title="FOMODEL2:def.21">DefTruth2</a></span>;<br/>
<a NAME="E9:59_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">C5</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K22" title="FOMODEL2:func.22">-TruthEval</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">K</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:59_1"><span class="lab"><font color="Green" title="E30">C3</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E10:59_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K23" title="FOMODEL2:func.23">-TruthEval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_1"><span class="lab"><font color="Green" title="E28">C2</font></span></a>, <a class="txt" href="#E7:59_1"><span class="lab"><font color="Green" title="E31">C8</font></span></a>, <a class="txt" href="#E2:59_1"><span class="lab"><font color="Green" title="E27">C1</font></span></a>, <a class="txt" href="#E6:59_1"><span class="lab"><font color="Green" title="E30">C3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K23" title="FOMODEL2:func.23">-TruthEval</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K23" title="FOMODEL2:func.23">-TruthEval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span><br/>

<a NAME="E11:59_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fomodel2.html#K19" title="FOMODEL2:func.19">ExIterator</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K22" title="FOMODEL2:func.22">-TruthEval</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">k</font></span>)</span></span>)</span> <a href="funct_4.html#K7" title="FUNCT_4:func.7">+*</a> <span class="p3">(<span class="default"><a href="fomodel2.html#K21" title="FOMODEL2:func.21">NorIterator</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K22" title="FOMODEL2:func.22">-TruthEval</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K22" title="FOMODEL2:func.22">-TruthEval</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">k</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:59_1"><span class="lab"><font color="Green" title="E32">C5</font></span></a>, <a class="txt" href="#E4:59_1"><span class="lab"><font color="Green" title="E29">C6</font></span></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:59_1"/>
<span class="p1">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K23" title="FOMODEL2:func.23">-TruthEval</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">U</font>) <a href="fomodel2.html#K23" title="FOMODEL2:func.23">-TruthEval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1"><span class="lab"><font color="Green" title="E27">C1</font></span></a>, <a class="txt" href="#E3:59_1"><span class="lab"><font color="Green" title="E28">C2</font></span></a>, <a class="txt" href="#E7:59_1"><span class="lab"><font color="Green" title="E31">C8</font></span></a>, <a class="txt" href="#E4:59_1"><span class="lab"><font color="Green" title="E29">C6</font></span></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>; <a class="txt" 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">thus </span><a NAME="E3:59"/>
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E25">B0</font></span></a>, <a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E26">B1</font></span></a>);</span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
