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<b>let </b><font color="Maroon" title="c1">A</font> be  <a href="aofa_i00.html#V2" title="AOFA_I00:attr.2">Euclidean</a> <a href="aofa_000.html#NM1" title="AOFA_000:NM.1">preIfWhileAlgebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">s</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Olive" title="b1">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> <br/> for <font color="Olive" title="b3">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">X</font><br/> for <font color="Olive" title="b4">g</font> being <a href="aofa_i00.html#V1" title="AOFA_I00:attr.1">Euclidean</a>  <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>, <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Olive" title="b1">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Olive" title="b1">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Olive" title="b3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <br/> for <font color="Olive" title="b5">I</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font><br/> for <font color="Olive" title="b6">i</font>, <font color="Olive" title="b7">n</font> being   <a href="aofa_i00.html#M1" title="AOFA_I00:mode.1">Variable</a> of <font color="Olive" title="b4">g</font>  st  ex <font color="Olive" title="b8">d</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b8">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b8">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b8">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) &amp; ( for <font color="Olive" title="b8">s</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Olive" title="b1">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  holds <br/> ( <span class="p1">(<span class="default"><font color="Olive" title="b4">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b8">s</font>,<font color="Olive" title="b5">I</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b7">n</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b4">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b8">s</font>,<font color="Olive" title="b5">I</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b6">i</font> ) ) holds <br/><font color="Olive" title="b4">g</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Olive" title="b6">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b7">n</font></span>)</span>,<font color="Olive" title="b4">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b2">s</font>,<span class="p1">(<span class="default"><font color="Olive" title="b6">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b7">n</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">s</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> <br/> for <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font><br/> for <font color="Olive" title="b3">g</font> being <a href="aofa_i00.html#V1" title="AOFA_I00:attr.1">Euclidean</a>  <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>, <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Olive" title="b2">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <br/> for <font color="Olive" title="b4">I</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font><br/> for <font color="Olive" title="b5">i</font>, <font color="Olive" title="b6">n</font> being   <a href="aofa_i00.html#M1" title="AOFA_I00:mode.1">Variable</a> of <font color="Olive" title="b3">g</font>  st  ex <font color="Olive" title="b7">d</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b7">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b7">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b7">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) &amp; ( for <font color="Olive" title="b7">s</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  holds <br/> ( <span class="p1">(<span class="default"><font color="Olive" title="b3">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b7">s</font>,<font color="Olive" title="b4">I</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b6">n</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b3">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b7">s</font>,<font color="Olive" title="b4">I</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b5">i</font> ) ) holds <br/><font color="Olive" title="b3">g</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Olive" title="b5">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b6">n</font></span>)</span>,<font color="Olive" title="b3">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><font color="Olive" title="b5">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b6">n</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">s</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font><br/> for <font color="Olive" title="b2">g</font> being <a href="aofa_i00.html#V1" title="AOFA_I00:attr.1">Euclidean</a>  <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>, <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Olive" title="b1">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <br/> for <font color="Olive" title="b3">I</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font><br/> for <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">n</font> being   <a href="aofa_i00.html#M1" title="AOFA_I00:mode.1">Variable</a> of <font color="Olive" title="b2">g</font>  st  ex <font color="Olive" title="b6">d</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b6">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b6">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b6">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) &amp; ( for <font color="Olive" title="b6">s</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  holds <br/> ( <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b6">s</font>,<font color="Olive" title="b3">I</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b5">n</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b6">s</font>,<font color="Olive" title="b3">I</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b4">i</font> ) ) holds <br/><font color="Olive" title="b2">g</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b5">n</font></span>)</span>,<font color="Olive" title="b2">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">s</font>,<span class="p1">(<span class="default"><font color="Olive" title="b4">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b5">n</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">g</font> being <a href="aofa_i00.html#V1" title="AOFA_I00:attr.1">Euclidean</a>  <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>, <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c4">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <br/> for <font color="Olive" title="b2">I</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font><br/> for <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">n</font> being   <a href="aofa_i00.html#M1" title="AOFA_I00:mode.1">Variable</a> of <font color="Olive" title="b1">g</font>  st  ex <font color="Olive" title="b5">d</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b5">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b5">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b5">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) &amp; ( for <font color="Olive" title="b5">s</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  holds <br/> ( <span class="p1">(<span class="default"><font color="Olive" title="b1">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b5">s</font>,<font color="Olive" title="b2">I</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b4">n</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b5">s</font>,<font color="Olive" title="b2">I</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b3">i</font> ) ) holds <br/><font color="Olive" title="b1">g</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b4">n</font></span>)</span>,<font color="Olive" title="b1">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">s</font>,<span class="p1">(<span class="default"><font color="Olive" title="b3">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b4">n</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c5">g</font> be  <a href="aofa_i00.html#V1" title="AOFA_I00:attr.1">Euclidean</a>  <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>, <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c4">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">I</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font><br/> for <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">n</font> being   <a href="aofa_i00.html#M1" title="AOFA_I00:mode.1">Variable</a> of <font color="Maroon" title="c5">g</font>  st  ex <font color="Olive" title="b4">d</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b4">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b4">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b4">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) &amp; ( for <font color="Olive" title="b4">s</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  holds <br/> ( <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b4">s</font>,<font color="Olive" title="b1">I</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b3">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b4">s</font>,<font color="Olive" title="b1">I</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b2">i</font> ) ) holds <br/><font color="Maroon" title="c5">g</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b3">n</font></span>)</span>,<font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">s</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b3">n</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c6">I</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">n</font> being   <a href="aofa_i00.html#M1" title="AOFA_I00:mode.1">Variable</a> of <font color="Maroon" title="c5">g</font>  st  ex <font color="Olive" title="b3">d</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b3">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b3">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b3">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) &amp; ( for <font color="Olive" title="b3">s</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  holds <br/> ( <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b3">s</font>,<font color="Maroon" title="c6">I</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b2">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b3">s</font>,<font color="Maroon" title="c6">I</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b1">i</font> ) ) holds <br/><font color="Maroon" title="c5">g</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b2">n</font></span>)</span>,<font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">s</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b2">n</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c7">i</font>, <font color="Maroon" title="c8">n</font> be    <a href="aofa_i00.html#M1" title="AOFA_I00:mode.1">Variable</a> of <font color="Maroon" title="c5">g</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">d</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b1">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) &amp; ( for <font color="Olive" title="b1">s</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  holds <br/> ( <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,<font color="Maroon" title="c6">I</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,<font color="Maroon" title="c6">I</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">i</font> ) ) implies <font color="Maroon" title="c5">g</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span>,<font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span> )</span><br/>













<b>given </b><font color="Maroon" title="c9">d</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><a NAME="E2:119"/><i><font color="Green" title="E64">A0</font></i>: 
( <font color="Maroon" title="c9">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c9">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c9">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">s</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,<font color="Maroon" title="c6">I</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">n</font> implies  not <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,<font color="Maroon" title="c6">I</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">i</font> ) or <font color="Maroon" title="c5">g</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span>,<font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span> )</span><br/>

<a NAME="E3:119"/><i><font color="Green" title="E65">D1</font></i>: 
( <font color="Maroon" title="c4">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">n</font> &amp; <font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">n</font> )
 
<b>by </b><i><a class="txt" href="#E2:119"><i><font color="Green" title="E64">A0</font></i></a></i>;<br/>
<b>assume </b><a NAME="E4:119"/><i><font color="Green" title="E66">00</font></i>: 
for <font color="Olive" title="b1">s</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  holds <br/> ( <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,<font color="Maroon" title="c6">I</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,<font color="Maroon" title="c6">I</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">i</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">g</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span>,<font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span></span><br/>

<b>set </b><font color="Maroon" title="c10">h</font> = <font color="Maroon" title="c5">g</font>;<br/>
<b>set </b><font color="Maroon" title="c11">S</font> =  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ;<br/>
<b>set </b><font color="Maroon" title="c12">T</font> = <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c4">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<b>set </b><font color="Maroon" title="c13">C</font> = <font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font>;<br/>
<b>set </b><font color="Maroon" title="c14">J</font> = <font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1;<br/>
<b>reconsider </b><font color="Maroon" title="c15">s1</font> = <font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ] <b>means </b>$1 <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">n</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  =  <a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">$1 <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default">$1 <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">i</font></span>)</span></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>
<a NAME="E6:119"/>
( ( <font color="Maroon" title="c3">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">n</font> implies <font color="Maroon" title="c15">s1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Maroon" title="c3">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">n</font> or <font color="Maroon" title="c3">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">n</font> ) &amp; <font color="Maroon" title="c15">s1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c15">s1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">n</font> &amp; ( <font color="Maroon" title="c3">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">n</font> implies <font color="Maroon" title="c15">s1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 
<b>by </b><i><a class="txt" href="#E2:119"><i><font color="Green" title="E64">A0</font></i></a>, <a class="ref" href="aofa_i00.html#T35" target="_self" title="AOFA_I00:th.35">Th114</a></i>;<br/>
<a NAME="E7:119"/><b>then </b><i><font color="Green" title="E67">AB</font></i>: 
( <font color="Maroon" title="c15">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c4">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">s1</font>] )
 
<b>by </b><i><a class="ref" href="aofa_i00.html#T2" target="_self" title="AOFA_I00:th.2">LemTS</a></i>;<br/>
<a NAME="E8:119"/><i><font color="Green" title="E68">C</font></i>: 
for <font color="Olive" title="b1">s</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">s</font>] holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span>] implies <font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c4">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c4">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span>] ) &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">s</font>) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">s</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c16">s</font>] implies ( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span>] implies <font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c4">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c4">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span>] ) &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">s</font>) ) )</span><br/>

<b>assume </b><a NAME="E1:119_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c16">s</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span>] implies <font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c4">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c4">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span>] ) &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">s</font>) )</span><br/>

<b>then reconsider </b><font color="Maroon" title="c17">ni</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c16">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">i</font></span>)</span> 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>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/>
<b>set </b><font color="Maroon" title="c18">s1</font> = <font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<font color="Maroon" title="c6">I</font>;<br/>
<b>set </b><font color="Maroon" title="c19">q</font> = <font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c20">q1</font> = <font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span>;<br/>
<a NAME="E3:119_1"/><i><font color="Green" title="E69">01</font></i>: 
( ( <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">n</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">n</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">n</font> ) &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">i</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">n</font> &amp; ( <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">n</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 
<b>by </b><i><a class="txt" href="#E2:119"><i><font color="Green" title="E64">A0</font></i></a>, <a class="ref" href="aofa_i00.html#T35" target="_self" title="AOFA_I00:th.35">Th114</a></i>;<br/>
<a NAME="E4:119_1"/><i><font color="Green" title="E70">02</font></i>: 
( <font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<font color="Maroon" title="c6">I</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span> &amp; <font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="aofa_000.html#D29" title="AOFA_000:def.29">AOFA_000:def 29</a></i>;<br/>
<b>hence </b><a NAME="E5:119_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span>] iff <font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c4">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="#E3:119_1"><i><font color="Green" title="E69">01</font></i></a>, <a class="ref" href="aofa_i00.html#T2" target="_self" title="AOFA_I00:th.2">LemTS</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">s</font>)</span><br/>

<a NAME="E6:119_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<font color="Maroon" title="c6">I</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">i</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<font color="Maroon" title="c6">I</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">n</font> )
 
<b>by </b><i><a class="txt" href="#E4:119"><i><font color="Green" title="E66">00</font></i></a></i>;<br/>
<a NAME="E7:119_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">n</font> )
 
<b>by </b><i><a class="txt" href="#E4:119_1"><i><font color="Green" title="E70">02</font></i></a>, <a class="txt" href="#E3:119"><i><font color="Green" title="E65">D1</font></i></a>, <a class="ref" href="aofa_i00.html#T28" target="_self" title="AOFA_I00:th.28">Th012</a></i>;<br/>
<a NAME="E8:119_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">n</font> )
 
<b>by </b><i><a class="txt" href="#E2:119"><i><font color="Green" title="E64">A0</font></i></a>, <a class="ref" href="aofa_i00.html#T35" target="_self" title="AOFA_I00:th.35">Th114</a></i>;<br/>
<a NAME="E9:119_1"/><b>then </b>
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">ni</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">s</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">ni</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></i>;<br/>
<b>hence </b><a NAME="E10:119_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">s</font>)
 <b>by </b><i><a class="txt" href="#E4:119_1"><i><font color="Green" title="E70">02</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E9:119"/>
<font color="Maroon" title="c5">g</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span>,<font color="Maroon" title="c15">s1</font>
 
<b>from </b><i><a class="ref" href="aofa_000.html#S3" title="AOFA_000:sch.3">AOFA_000:sch 3</a>(<a class="txt" href="#E7:119"><i><font color="Green" title="E67">AB</font></i></a>, <a class="txt" href="#E8:119"><i><font color="Green" title="E68">C</font></i></a>);<br/></i>
<b>hence </b><a NAME="E10:119"/>
<font color="Maroon" title="c5">g</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span>,<font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c8">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
