<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">A</font> be  <a href="aofa_i00.html#V2" title="AOFA_I00:attr.2">Euclidean</a> <a href="aofa_000.html#NM1" title="AOFA_000:NM.1">preIfWhileAlgebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">s</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Olive" title="b1">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> <br/> for <font color="Olive" title="b3">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">X</font><br/> for <font color="Olive" title="b4">g</font> being <a href="aofa_i00.html#V1" title="AOFA_I00:attr.1">Euclidean</a>  <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>, <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Olive" title="b1">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Olive" title="b1">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Olive" title="b3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <br/> for <font color="Olive" title="b5">x</font>, <font color="Olive" title="b6">y</font> being   <a href="aofa_i00.html#M1" title="AOFA_I00:mode.1">Variable</a> of <font color="Olive" title="b4">g</font> holds <br/> ( ( <font color="Olive" title="b2">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b6">y</font> implies <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="b2">s</font>,<span class="p2">(<span class="default"><font color="Olive" title="b5">x</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b6">y</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; ( <font color="Olive" title="b2">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b5">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b6">y</font> implies <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="b2">s</font>,<span class="p2">(<span class="default"><font color="Olive" title="b5">x</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b6">y</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</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; ( for <font color="Olive" title="b7">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">X</font>  st <font color="Olive" title="b7">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">b</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="b2">s</font>,<span class="p2">(<span class="default"><font color="Olive" title="b5">x</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b6">y</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b7">z</font> ) )</span><br/><b>let </b><font color="Maroon" title="c2">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">s</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> <br/> for <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font><br/> for <font color="Olive" title="b3">g</font> being <a href="aofa_i00.html#V1" title="AOFA_I00:attr.1">Euclidean</a>  <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>, <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Olive" title="b2">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <br/> for <font color="Olive" title="b4">x</font>, <font color="Olive" title="b5">y</font> being   <a href="aofa_i00.html#M1" title="AOFA_I00:mode.1">Variable</a> of <font color="Olive" title="b3">g</font> holds <br/> ( ( <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b4">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b5">y</font> implies <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="b1">s</font>,<span class="p2">(<span class="default"><font color="Olive" title="b4">x</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b5">y</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; ( <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b4">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b5">y</font> implies <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="b1">s</font>,<span class="p2">(<span class="default"><font color="Olive" title="b4">x</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b5">y</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</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; ( for <font color="Olive" title="b6">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b6">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</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="b1">s</font>,<span class="p2">(<span class="default"><font color="Olive" title="b4">x</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b5">y</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b6">z</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="Olive" title="b6">z</font> ) )</span><br/><b>let </b><font color="Maroon" title="c3">s</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font><br/> for <font color="Olive" title="b2">g</font> being <a href="aofa_i00.html#V1" title="AOFA_I00:attr.1">Euclidean</a>  <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>, <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Olive" title="b1">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <br/> for <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being   <a href="aofa_i00.html#M1" title="AOFA_I00:mode.1">Variable</a> of <font color="Olive" title="b2">g</font> holds <br/> ( ( <font color="Maroon" title="c3">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b3">x</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="Olive" title="b4">y</font> implies <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="Maroon" title="c3">s</font>,<span class="p2">(<span class="default"><font color="Olive" title="b3">x</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b4">y</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; ( <font color="Maroon" title="c3">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b3">x</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="Olive" title="b4">y</font> implies <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="Maroon" title="c3">s</font>,<span class="p2">(<span class="default"><font color="Olive" title="b3">x</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b4">y</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</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; ( for <font color="Olive" title="b5">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b5">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">b</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="Maroon" title="c3">s</font>,<span class="p2">(<span class="default"><font color="Olive" title="b3">x</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b4">y</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b5">z</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="Olive" title="b5">z</font> ) )</span><br/><b>let </b><font color="Maroon" title="c4">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">g</font> being <a href="aofa_i00.html#V1" title="AOFA_I00:attr.1">Euclidean</a>  <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>, <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c4">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <br/> for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="aofa_i00.html#M1" title="AOFA_I00:mode.1">Variable</a> of <font color="Olive" title="b1">g</font> holds <br/> ( ( <font color="Maroon" title="c3">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b2">x</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="Olive" title="b3">y</font> implies <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="Maroon" title="c3">s</font>,<span class="p2">(<span class="default"><font color="Olive" title="b2">x</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b3">y</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 ) &amp; ( <font color="Maroon" title="c3">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b2">x</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="Olive" title="b3">y</font> implies <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="Maroon" title="c3">s</font>,<span class="p2">(<span class="default"><font color="Olive" title="b2">x</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b3">y</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( for <font color="Olive" title="b4">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b4">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</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="Maroon" title="c3">s</font>,<span class="p2">(<span class="default"><font color="Olive" title="b2">x</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b3">y</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b4">z</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="Olive" title="b4">z</font> ) )</span><br/><b>let </b><font color="Maroon" title="c5">f</font> be  <a href="aofa_i00.html#V1" title="AOFA_I00:attr.1">Euclidean</a>  <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>, <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c4">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="aofa_i00.html#M1" title="AOFA_I00:mode.1">Variable</a> of <font color="Maroon" title="c5">f</font> holds <br/> ( ( <font color="Maroon" title="c3">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b1">x</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="Olive" title="b2">y</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">s</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b2">y</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 ) &amp; ( <font color="Maroon" title="c3">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b1">x</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="Olive" title="b2">y</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">s</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b2">y</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( for <font color="Olive" title="b3">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b3">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">s</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b2">y</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b3">z</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="Olive" title="b3">z</font> ) )</span><br/><b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be    <a href="aofa_i00.html#M1" title="AOFA_I00:mode.1">Variable</a> of <font color="Maroon" title="c5">f</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c3">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c6">x</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="c7">y</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c7">y</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 ) &amp; ( <font color="Maroon" title="c3">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c6">x</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="c7">y</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( for <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b1">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b1">z</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="Olive" title="b1">z</font> ) )</span><br/>











<b>reconsider </b><font color="Maroon" title="c8">b'</font> = <font color="Maroon" title="c4">b</font> as    <a href="aofa_i00.html#M1" title="AOFA_I00:mode.1">Variable</a> of <font color="Maroon" title="c5">f</font> <b>by </b><i><a class="ref" href="aofa_i00.html#D2" target="_self" title="AOFA_I00:def.2">ELEM</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">x'</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c10">y'</font> = <font color="Maroon" title="c7">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> ;<br/>
<b>set </b><font color="Maroon" title="c11">v</font> =  <a href="aofa_i00.html#NK43" title="AOFA_I00:NK.43">^</a> <font color="Maroon" title="c8">b'</font>;<br/>
<b>set </b><font color="Maroon" title="c12">t</font> =  <a href="aofa_i00.html#K37" title="AOFA_I00:func.37">leq</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c6">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">y</font></span>)</span>;<br/>
<a NAME="E3:100"/>
( <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="aofa_i00.html#NK43" title="AOFA_I00:NK.43">^</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font></span>)</span> &amp;  <a href="aofa_i00.html#NK43" title="AOFA_I00:NK.43">^</a> <font color="Maroon" title="c6">x</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c6">x</font> &amp; <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="aofa_i00.html#NK43" title="AOFA_I00:NK.43">^</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font></span>)</span> &amp;  <a href="aofa_i00.html#NK43" title="AOFA_I00:NK.43">^</a> <font color="Maroon" title="c7">y</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c7">y</font> )
 
<b>by </b><i><a class="ref" href="aofa_i00.html#D19" target="_self" title="AOFA_I00:def.19">DEFvarexp</a></i>;<br/>
<a NAME="E4:100"/><b>then </b><i><font color="Green" title="E53">01</font></i>: 
( <font color="Maroon" title="c3">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font> &amp; <font color="Maroon" title="c3">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font> &amp; <span class="p1">(<span class="default"><a href="aofa_i00.html#NK43" title="AOFA_I00:NK.43">^</a> <font color="Maroon" title="c8">b'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E5:100"/><i><font color="Green" title="E54">02</font></i>: 
( <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font> implies  <a href="xxreal_0.html#K5" title="XXREAL_0:func.5">IFGT</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#D10" title="XXREAL_0:def.10">XXREAL_0:def 10</a></i>;<br/>
<a NAME="E6:100"/><i><font color="Green" title="E55">03</font></i>: 
( <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font> implies  <a href="xxreal_0.html#K5" title="XXREAL_0:func.5">IFGT</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#D10" title="XXREAL_0:def.10">XXREAL_0:def 10</a></i>;<br/>
<a NAME="E7:100"/>
( <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="aofa_i00.html#K67" title="AOFA_I00:func.67">leq</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#NK43" title="AOFA_I00:NK.43">^</a> <font color="Maroon" title="c8">b'</font></span>)</span> <a href="aofa_i00.html#K44" title="AOFA_I00:func.44">:=</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K37" title="AOFA_I00:func.37">leq</a> <span class="p2">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c6">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="aofa_i00.html#K37" title="AOFA_I00:func.37">leq</a> <span class="p2">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c6">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K5" title="XXREAL_0:func.5">IFGT</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 )
 
<b>by </b><i><a class="ref" href="aofa_i00.html#D31" target="_self" title="AOFA_I00:def.31">DEFleq2</a></i>;<br/>
<b>hence </b><a NAME="E8:100"/>
( ( <font color="Maroon" title="c3">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c6">x</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="c7">y</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c7">y</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 ) &amp; ( <font color="Maroon" title="c3">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c6">x</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="c7">y</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( for <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b1">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b1">z</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="Olive" title="b1">z</font> ) )
 <b>by </b><i><a class="txt" href="aofa_i00.html#E4:100"><i><font color="Green" title="E53">01</font></i></a>, <a class="txt" href="aofa_i00.html#E5:100"><i><font color="Green" title="E54">02</font></i></a>, <a class="txt" href="aofa_i00.html#E6:100"><i><font color="Green" title="E55">03</font></i></a>, <a class="ref" href="aofa_i00.html#T24" target="_self" title="AOFA_I00:th.24">Th100</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
