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<b>let </b><font color="Maroon" title="c1">A</font> be   <a href="aofa_000.html#NM1" title="AOFA_000:NM.1">preIfWhileAlgebra</a>;<br/><b>let </b><font color="Maroon" title="c2">I</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>;<br/><b>let </b><font color="Maroon" title="c3">S</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> ;<br/><b>let </b><font color="Maroon" title="c4">T</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">S</font>;<br/><b>let </b><font color="Maroon" title="c5">s</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">S</font>;<br/><b>let </b><font color="Maroon" title="c6">f</font> be    <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font>;<br/>











<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:141_1"/>
(  not <font color="Maroon" title="c6">f</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c5">s</font> or <font color="Maroon" title="c6">f</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c5">s</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_1_1"><b>suppose </b></a><a NAME="E1:141_1_1"/>
not <font color="Maroon" title="c6">f</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c5">s</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:141_1_1"/>
 <a href="aofa_000.html#K20" title="AOFA_000:func.20">iteration-degree</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c6">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="aofa_000.html#D34" target="_self" title="AOFA_000:def.34">Def34</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_1_2"><b>suppose </b></a><a NAME="E1:141_1_2"/>
<font color="Maroon" title="c6">f</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c5">s</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">r</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">S</font><b> such that </b><br/><a NAME="E3:141_1_2"/><i><font color="Green" title="E90">A1</font></i>: 
 <a href="aofa_000.html#K20" title="AOFA_000:func.20">iteration-degree</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>and </b><br/><a NAME="E4:141_1_2"/>
( <font color="Maroon" title="c7">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> &amp; <font color="Maroon" title="c7">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c4">T</font> &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">r</font> holds <br/>( <font color="Maroon" title="c7">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">T</font> &amp; <font color="Maroon" title="c7">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</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> <font color="Maroon" title="c6">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<font color="Maroon" title="c2">I</font> ) ) )
 <b>by </b><i><a class="ref" href="aofa_000.html#D34" target="_self" title="AOFA_000:def.34">Def34</a></i>;<br/><a NAME="E5:141_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/><a NAME="E6:141_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/><b>hence </b><a NAME="E7:141_1_2"/>
 <a href="aofa_000.html#K20" title="AOFA_000:func.20">iteration-degree</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c6">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="aofa_000.html#E3:141_1_2"><i><font color="Green" title="E90">A1</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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