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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Olive" title="b1">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">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="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Olive" title="b1">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Olive" title="b1">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">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">P</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b6">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="b7">i</font>, <font color="Olive" title="b8">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="b9">d</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b9">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="b9">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b9">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) &amp; ( for <font color="Olive" title="b9">s</font> being   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Olive" title="b1">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>   st <font color="Olive" title="b9">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">P</font> 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="b9">s</font>,<font color="Olive" title="b6">I</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b9">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b8">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="b9">s</font>,<font color="Olive" title="b6">I</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b9">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b7">i</font> &amp; <font color="Olive" title="b4">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b9">s</font>,<font color="Olive" title="b6">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">P</font> &amp; <font color="Olive" title="b4">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b9">s</font>,<span class="p1">(<span class="default"><font color="Olive" title="b7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b8">n</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">P</font> &amp; <font color="Olive" title="b4">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b9">s</font>,<span class="p1">(<span class="default"><font color="Olive" title="b7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">P</font> ) ) &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">P</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="b6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Olive" title="b7">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="b7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b8">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="b7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b8">n</font></span>)</span></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">s</font> being   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">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="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">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">P</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="b3">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="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="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="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>   st <font color="Olive" title="b8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">P</font> 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="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="b3">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> &amp; <font color="Olive" title="b3">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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">P</font> &amp; <font color="Olive" title="b3">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b8">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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">P</font> &amp; <font color="Olive" title="b3">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b8">s</font>,<span class="p1">(<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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">P</font> ) ) &amp; <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">P</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="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="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="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/><span class="kw">let </span><font color="Maroon" title="c3">s</font> be    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">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">P</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="b2">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="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="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="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>   st <font color="Olive" title="b7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">P</font> 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="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="b2">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> &amp; <font color="Olive" title="b2">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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">P</font> &amp; <font color="Olive" title="b2">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b7">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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">P</font> &amp; <font color="Olive" title="b2">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b7">s</font>,<span class="p1">(<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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">P</font> ) ) &amp; <font color="Maroon" title="c3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">P</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="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="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="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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">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">P</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="b1">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="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="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="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>   st <font color="Olive" title="b6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">P</font> 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="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="b1">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> &amp; <font color="Olive" title="b1">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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">P</font> &amp; <font color="Olive" title="b1">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b6">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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">P</font> &amp; <font color="Olive" title="b1">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b6">s</font>,<span class="p1">(<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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">P</font> ) ) &amp; <font color="Maroon" title="c3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">P</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="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="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="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/><span class="kw">let </span><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="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">P</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="Maroon" title="c5">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="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>   st <font color="Olive" title="b5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> 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="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="Maroon" title="c5">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> &amp; <font color="Maroon" title="c5">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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b5">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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b5">s</font>,<span class="p1">(<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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> ) ) &amp; <font color="Maroon" title="c3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</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="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="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="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/><span class="kw">let </span><font color="Maroon" title="c6">P</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>   st <font color="Olive" title="b4">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> 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> &amp; <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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <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>,<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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <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>,<span class="p1">(<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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> ) ) &amp; <font color="Maroon" title="c3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</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/><span class="kw">let </span><font color="Maroon" title="c7">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>   st <font color="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> 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="c7">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="c7">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> &amp; <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="c7">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <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>,<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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <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>,<span class="p1">(<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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> ) ) &amp; <font color="Maroon" title="c3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</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="c7">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/><span class="kw">let </span><font color="Maroon" title="c8">i</font>, <font color="Maroon" title="c9">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c9">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="c8">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="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>   st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> 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="c7">I</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">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="c9">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="c7">I</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">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="c8">i</font> &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>,<font color="Maroon" title="c7">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> &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"><font color="Maroon" title="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> &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"><font color="Maroon" title="c8">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> ) ) &amp; <font color="Maroon" title="c3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</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="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">n</font></span>)</span> )</span><br/>















<span class="kw">given </span><font color="Maroon" title="c10">d</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><a NAME="E2:120"/><span class="lab"><font color="Green" title="E64">A1</font></span>: 
<font color="Maroon" title="c10">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> 
 <span class="kw">and </span><br/><a NAME="E3:120"/><span class="lab"><font color="Green" title="E65">A2</font></span>: 
<font color="Maroon" title="c10">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">and </span><br/><a NAME="E4:120"/><span class="lab"><font color="Green" title="E66">A3</font></span>: 
<font color="Maroon" title="c10">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">s</font> being   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  st <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; not <!--HUMANRECFAILED-->( <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="c7">I</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">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="c9">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="c7">I</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">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="c8">i</font> &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>,<font color="Maroon" title="c7">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> &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"><font color="Maroon" title="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> &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"><font color="Maroon" title="c8">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> ) ) or  not <font color="Maroon" title="c3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</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="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">n</font></span>)</span> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c11">J</font> = <font color="Maroon" title="c8">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">C</font> = <font color="Maroon" title="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">n</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">S</font> =  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">h</font> = <font color="Maroon" title="c5">g</font>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E5:120"/><span class="lab"><font color="Green" title="E67">A4</font></span>: 
for <font color="Olive" title="b1">s</font> being   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>   st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> 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="c7">I</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">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="c9">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="c7">I</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">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="c8">i</font> &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>,<font color="Maroon" title="c7">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> &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"><font color="Maroon" title="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> &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"><font color="Maroon" title="c8">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> )
 <span class="kw">and </span><br/><a NAME="E6:120"/><span class="lab"><font color="Green" title="E68">A5</font></span>: 
<font color="Maroon" title="c3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">n</font></span>)</span></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ] <span class="kw">means </span>$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font>;<br/>
<span class="kw">reconsider </span><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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">n</font></span>)</span> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  ;<br/>
<a NAME="E8:120"/><span class="lab"><font color="Green" title="E69">A6</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">s1</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:120"><span class="lab"><font color="Green" title="E67">A4</font></span></a>, <a class="txt" href="#E6:120"><span class="lab"><font color="Green" title="E68">A5</font></span></a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ) <span class="kw">-&gt; </span>   <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="c9">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="c8">i</font></span>)</span></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ] <span class="kw">means </span>$1 <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">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="c9">n</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">T</font> = <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">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/>
<a NAME="E9:120"/><span class="lab"><font color="Green" title="E70">A7</font></span>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:120"><span class="lab"><font color="Green" title="E65">A2</font></span></a>, <a class="txt" href="#E4:120"><span class="lab"><font color="Green" title="E66">A3</font></span></a></span>;<br/>
<a NAME="E10:120"/><span class="lab"><font color="Green" title="E71">A8</font></span>: 
for <font color="Olive" title="b1">s</font> being   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">s</font>] &amp; <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">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">S<sub>2</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="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span>] &amp; ( <font color="Maroon">S<sub>2</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="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">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="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">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="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">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="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">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="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">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>2</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="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">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="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">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="120_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">s</font> be    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c17">s</font>] &amp; <font color="Maroon" title="c17">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">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">S<sub>2</sub></font>[<font color="Maroon" title="c17">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="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span>] &amp; ( <font color="Maroon">S<sub>2</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="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">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="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">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="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">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="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">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="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">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>2</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="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">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="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">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="c17">s</font>) ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:120_1"/><span class="lab"><font color="Green" title="E72">A9</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c17">s</font>]
 <span class="kw">and </span><br/><a NAME="E2:120_1"/>
<font color="Maroon" title="c17">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">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> 
 <span class="kw">and </span><br/><a NAME="E3:120_1"/><span class="lab"><font color="Green" title="E73">A10</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c17">s</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span>] &amp; ( <font color="Maroon">S<sub>2</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="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">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="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">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="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">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="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">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="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">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>2</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="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">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="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">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="c17">s</font>) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c18">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="c17">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c19">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="c17">s</font>,<font color="Maroon" title="c7">I</font>;<br/>
<span class="kw">set </span><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="c17">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">n</font></span>)</span>;<br/>
<a NAME="E4:120_1"/><span class="lab"><font color="Green" title="E74">A11</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">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="c17">s</font>,<font color="Maroon" title="c7">I</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_000.html#D29" title="AOFA_000:def.29">AOFA_000:def 29</a></span>;<br/>
<a NAME="E5:120_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="c17">s</font>,<font color="Maroon" title="c7">I</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:120"><span class="lab"><font color="Green" title="E67">A4</font></span></a>, <a class="txt" href="#E1:120_1"><span class="lab"><font color="Green" title="E72">A9</font></span></a></span>;<br/>
<a NAME="E6:120_1"/><span class="kw">then </span>
<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="c17">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:120_1"><span class="lab"><font color="Green" title="E74">A11</font></span></a>, <a class="ref" href="aofa_i00.html#T28" target="_self" title="AOFA_I00:th.28">Th28</a></span>;<br/>
<a NAME="E7:120_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A12</font></span>: 
<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="c17">s</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:120"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="txt" href="#E4:120"><span class="lab"><font color="Green" title="E66">A3</font></span></a>, <a class="ref" href="aofa_i00.html#T35" target="_self" title="AOFA_I00:th.35">Th35</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">ni</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c17">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">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="c17">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:120_1"><span class="lab"><font color="Green" title="E73">A10</font></span></a>, <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></span>;<br/>
<a NAME="E9:120_1"/><span class="lab"><font color="Green" title="E76">A13</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">s</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">ni</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></span>;<br/>
<a NAME="E10:120_1"/><span class="lab"><font color="Green" title="E77">A14</font></span>: 
( <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="c17">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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">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="c17">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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="c9">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="c17">s</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">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 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_i00.html#T35" target="_self" title="AOFA_I00:th.35">Th35</a></span>;<br/>
<a NAME="E11:120_1"/><span class="lab"><font color="Green" title="E78">A15</font></span>: 
<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="c17">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">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="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_000.html#D29" title="AOFA_000:def.29">AOFA_000:def 29</a></span>;<br/>
<a NAME="E12:120_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="c17">s</font>,<font color="Maroon" title="c7">I</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:120"><span class="lab"><font color="Green" title="E67">A4</font></span></a>, <a class="txt" href="#E1:120_1"><span class="lab"><font color="Green" title="E72">A9</font></span></a></span>;<br/>
<a NAME="E13:120_1"/><span class="kw">then </span>
<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="c17">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:120"><span class="lab"><font color="Green" title="E67">A4</font></span></a>, <a class="txt" href="#E4:120_1"><span class="lab"><font color="Green" title="E74">A11</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:120_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="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:120"><span class="lab"><font color="Green" title="E67">A4</font></span></a>, <a class="txt" href="#E11:120_1"><span class="lab"><font color="Green" title="E78">A15</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon">S<sub>2</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="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">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="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">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="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">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="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">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="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">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>2</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="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">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="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">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="c17">s</font>) )</span><br/>

<a NAME="E15:120_1"/><span class="lab"><font color="Green" title="E79">A16</font></span>: 
( <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="c17">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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">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="c17">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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="c9">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="c17">s</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">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>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_i00.html#T35" target="_self" title="AOFA_I00:th.35">Th35</a></span>;<br/>
<a NAME="E16:120_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> <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="c17">s</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">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="c17">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:120"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="txt" href="#E4:120"><span class="lab"><font color="Green" title="E66">A3</font></span></a>, <a class="ref" href="aofa_i00.html#T35" target="_self" title="AOFA_I00:th.35">Th35</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:120_1"/>
( <font color="Maroon">S<sub>2</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="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">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="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">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="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:120"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="txt" href="#E3:120"><span class="lab"><font color="Green" title="E65">A2</font></span></a>, <a class="txt" href="#E15:120_1"><span class="lab"><font color="Green" title="E79">A16</font></span></a>, <a class="txt" href="#E10:120_1"><span class="lab"><font color="Green" title="E77">A14</font></span></a>, <a class="txt" href="#E11:120_1"><span class="lab"><font color="Green" title="E78">A15</font></span></a>, <a class="ref" href="aofa_i00.html#T2" target="_self" title="AOFA_I00:th.2">Th2</a>, <a class="ref" href="aofa_i00.html#T35" target="_self" title="AOFA_I00:th.35">Th35</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">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="c17">s</font>)</span><br/>

<a NAME="E18:120_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="c17">s</font>,<font color="Maroon" title="c7">I</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:120"><span class="lab"><font color="Green" title="E67">A4</font></span></a>, <a class="txt" href="#E1:120_1"><span class="lab"><font color="Green" title="E72">A9</font></span></a></span>;<br/>
<a NAME="E19:120_1"/><span class="kw">then </span>
<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="c17">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:120"><span class="lab"><font color="Green" title="E70">A7</font></span></a>, <a class="txt" href="#E4:120_1"><span class="lab"><font color="Green" title="E74">A11</font></span></a>, <a class="ref" href="aofa_i00.html#T28" target="_self" title="AOFA_I00:th.28">Th28</a></span>;<br/>
<a NAME="E20:120_1"/><span class="kw">then </span>
<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="c17">s</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:120"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="txt" href="#E3:120"><span class="lab"><font color="Green" title="E65">A2</font></span></a>, <a class="ref" href="aofa_i00.html#T35" target="_self" title="AOFA_I00:th.35">Th35</a></span>;<br/>
<a NAME="E21:120_1"/><span class="kw">then </span>
<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="c17">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">n</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">ni</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:120_1"><span class="lab"><font color="Green" title="E75">A12</font></span></a>, <a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E22:120_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="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">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="c17">s</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:120_1"><span class="lab"><font color="Green" title="E78">A15</font></span></a>, <a class="txt" href="#E9:120_1"><span class="lab"><font color="Green" title="E76">A13</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E11:120"/><span class="lab"><font color="Green" title="E72">A17</font></span>: 
( <font color="Maroon" title="c3">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">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="c9">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>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_i00.html#T35" target="_self" title="AOFA_I00:th.35">Th35</a></span>;<br/>
<a NAME="E12:120"/><span class="lab"><font color="Green" title="E73">A18</font></span>: 
( <font color="Maroon" title="c3">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">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="c9">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 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_i00.html#T35" target="_self" title="AOFA_I00:th.35">Th35</a></span>;<br/>
<a NAME="E13:120"/>
<font color="Maroon" title="c15">s1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">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="c8">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:120"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="txt" href="#E4:120"><span class="lab"><font color="Green" title="E66">A3</font></span></a>, <a class="ref" href="aofa_i00.html#T35" target="_self" title="AOFA_I00:th.35">Th35</a></span>;<br/>
<a NAME="E14:120"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A19</font></span>: 
( <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="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">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>2</sub></font>[<font color="Maroon" title="c15">s1</font>] )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:120"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="txt" href="#E3:120"><span class="lab"><font color="Green" title="E65">A2</font></span></a>, <a class="txt" href="#E11:120"><span class="lab"><font color="Green" title="E72">A17</font></span></a>, <a class="txt" href="#E12:120"><span class="lab"><font color="Green" title="E73">A18</font></span></a>, <a class="ref" href="aofa_i00.html#T2" target="_self" title="AOFA_I00:th.2">Th2</a>, <a class="ref" href="aofa_i00.html#T35" target="_self" title="AOFA_I00:th.35">Th35</a></span>;<br/>
<a NAME="E15:120"/>
<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="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">n</font></span>)</span>,<font color="Maroon" title="c15">s1</font>
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="aofa_000.html#S4" title="AOFA_000:sch.4">AOFA_000:sch 4</a>(<a class="txt" href="#E8:120"><span class="lab"><font color="Green" title="E69">A6</font></span></a>, <a class="txt" href="#E14:120"><span class="lab"><font color="Green" title="E74">A19</font></span></a>, <a class="txt" href="#E10:120"><span class="lab"><font color="Green" title="E71">A8</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E16:120"/>
<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="c7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">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="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c9">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
