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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">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#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#K75" title="AOFA_I00:func.75">gt</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#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#K75" title="AOFA_I00:func.75">gt</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; ( <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#NR3" title="XXREAL_0:NR.3">&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#NK83" title="AOFA_I00:NK.83">lt</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#NR2" title="XXREAL_0:NR.2">&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#NK83" title="AOFA_I00:NK.83">lt</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#K75" title="AOFA_I00:func.75">gt</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> &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="b2">s</font>,<span class="p2">(<span class="default"><font color="Olive" title="b5">x</font> <a href="aofa_i00.html#NK83" title="AOFA_I00:NK.83">lt</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/><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">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#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#K75" title="AOFA_I00:func.75">gt</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#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#K75" title="AOFA_I00:func.75">gt</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; ( <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#NR3" title="XXREAL_0:NR.3">&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#NK83" title="AOFA_I00:NK.83">lt</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#NR2" title="XXREAL_0:NR.2">&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#NK83" title="AOFA_I00:NK.83">lt</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#K75" title="AOFA_I00:func.75">gt</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> &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="b1">s</font>,<span class="p2">(<span class="default"><font color="Olive" title="b4">x</font> <a href="aofa_i00.html#NK83" title="AOFA_I00:NK.83">lt</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/><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">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#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#K75" title="AOFA_I00:func.75">gt</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#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#K75" title="AOFA_I00:func.75">gt</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; ( <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#NR3" title="XXREAL_0:NR.3">&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#NK83" title="AOFA_I00:NK.83">lt</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#NR2" title="XXREAL_0:NR.2">&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#NK83" title="AOFA_I00:NK.83">lt</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#K75" title="AOFA_I00:func.75">gt</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> &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="Maroon" title="c3">s</font>,<span class="p2">(<span class="default"><font color="Olive" title="b3">x</font> <a href="aofa_i00.html#NK83" title="AOFA_I00:NK.83">lt</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/><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">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#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#K75" title="AOFA_I00:func.75">gt</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#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#K75" title="AOFA_I00:func.75">gt</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; ( <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#NR3" title="XXREAL_0:NR.3">&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#NK83" title="AOFA_I00:NK.83">lt</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#NR2" title="XXREAL_0:NR.2">&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#NK83" title="AOFA_I00:NK.83">lt</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#K75" title="AOFA_I00:func.75">gt</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> &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="Maroon" title="c3">s</font>,<span class="p2">(<span class="default"><font color="Olive" title="b2">x</font> <a href="aofa_i00.html#NK83" title="AOFA_I00:NK.83">lt</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/><span class="kw">let </span><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="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">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#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#K75" title="AOFA_I00:func.75">gt</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#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#K75" title="AOFA_I00:func.75">gt</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; ( <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#NR3" title="XXREAL_0:NR.3">&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#NK83" title="AOFA_I00:NK.83">lt</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#NR2" title="XXREAL_0:NR.2">&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#NK83" title="AOFA_I00:NK.83">lt</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#K75" title="AOFA_I00:func.75">gt</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> &amp; <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#NK83" title="AOFA_I00:NK.83">lt</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/>









<span class="kw">reconsider </span><font color="Maroon" title="c6">b9</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> <span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_i00.html#D2" target="_self" title="AOFA_I00:def.2">Def2</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c7">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="c8">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="c7">x</font> <a href="aofa_i00.html#K75" title="AOFA_I00:func.75">gt</a> <font color="Maroon" title="c8">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="c7">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="c8">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="c7">x</font> <a href="aofa_i00.html#K75" title="AOFA_I00:func.75">gt</a> <font color="Maroon" title="c8">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; ( <font color="Maroon" title="c3">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">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="c7">x</font> <a href="aofa_i00.html#NK83" title="AOFA_I00:NK.83">lt</a> <font color="Maroon" title="c8">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="c7">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">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="c7">x</font> <a href="aofa_i00.html#NK83" title="AOFA_I00:NK.83">lt</a> <font color="Maroon" title="c8">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="c7">x</font> <a href="aofa_i00.html#K75" title="AOFA_I00:func.75">gt</a> <font color="Maroon" title="c8">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> &amp; <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="c7">x</font> <a href="aofa_i00.html#NK83" title="AOFA_I00:NK.83">lt</a> <font color="Maroon" title="c8">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/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">x9</font> = <font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c10">y9</font> = <font color="Maroon" title="c8">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/>
<span class="kw">set </span><font color="Maroon" title="c11">v</font> =  <a href="aofa_i00.html#NK43" title="AOFA_I00:NK.43">^</a> <font color="Maroon" title="c6">b9</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">t</font> =  <a href="aofa_i00.html#K38" title="AOFA_I00:func.38">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">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="c8">y</font></span>)</span>;<br/>
<a NAME="E3:104"/><span class="lab"><font color="Green" title="E54">A1</font></span>: 
<span class="p1">(<span class="default"><a href="aofa_i00.html#NK43" title="AOFA_I00:NK.43">^</a> <font color="Maroon" title="c6">b9</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<a NAME="E4:104"/><span class="lab"><font color="Green" title="E55">A2</font></span>: 
( <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">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="c8">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="c7">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="c8">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>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D11" title="XXREAL_0:def.11">XXREAL_0:def 11</a></span>;<br/>
<a NAME="E5:104"/><span class="lab"><font color="Green" title="E56">A3</font></span>: 
<span class="p1">(<span class="default"><a href="aofa_i00.html#K38" title="AOFA_I00:func.38">gt</a> <span class="p2">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">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="c8">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="c7">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="c8">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>,1,<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#D32" target="_self" title="AOFA_I00:def.32">Def32</a></span>;<br/>
<a NAME="E6:104"/><span class="lab"><font color="Green" title="E57">A4</font></span>: 
( <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">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#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">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="c8">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>,<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">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>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D11" title="XXREAL_0:def.11">XXREAL_0:def 11</a></span>;<br/>
<a NAME="E7:104"/>
<span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">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="c7">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="kw">by</span> <span class="lab"><a class="ref" href="aofa_i00.html#D19" target="_self" title="AOFA_I00:def.19">Def19</a></span>;<br/>
<a NAME="E8:104"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A5</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="c7">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="c7">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<a NAME="E9:104"/><span class="lab"><font color="Green" title="E59">A6</font></span>: 
( <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">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="c8">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="c7">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="c8">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>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="xxreal_0.html#D11" title="XXREAL_0:def.11">XXREAL_0:def 11</a></span>;<br/>
<a NAME="E10:104"/><span class="lab"><font color="Green" title="E60">A7</font></span>: 
<span class="p1">(<span class="default"><a href="aofa_i00.html#K38" title="AOFA_I00:func.38">gt</a> <span class="p2">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</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">x</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="c8">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>,<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">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>,1,<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#D32" target="_self" title="AOFA_I00:def.32">Def32</a></span>;<br/>
<a NAME="E11:104"/><span class="lab"><font color="Green" title="E61">A8</font></span>: 
( <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">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#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">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="c8">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>,<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">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>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="xxreal_0.html#D11" title="XXREAL_0:def.11">XXREAL_0:def 11</a></span>;<br/>
<a NAME="E12:104"/>
<span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">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="c8">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_i00.html#D19" target="_self" title="AOFA_I00:def.19">Def19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:104"/>
( ( <font color="Maroon" title="c3">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">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="c8">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="c7">x</font> <a href="aofa_i00.html#K75" title="AOFA_I00:func.75">gt</a> <font color="Maroon" title="c8">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="c7">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="c8">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="c7">x</font> <a href="aofa_i00.html#K75" title="AOFA_I00:func.75">gt</a> <font color="Maroon" title="c8">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; ( <font color="Maroon" title="c3">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">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="c7">x</font> <a href="aofa_i00.html#NK83" title="AOFA_I00:NK.83">lt</a> <font color="Maroon" title="c8">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="c7">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">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="c7">x</font> <a href="aofa_i00.html#NK83" title="AOFA_I00:NK.83">lt</a> <font color="Maroon" title="c8">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="c7">x</font> <a href="aofa_i00.html#K75" title="AOFA_I00:func.75">gt</a> <font color="Maroon" title="c8">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> &amp; <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="c7">x</font> <a href="aofa_i00.html#NK83" title="AOFA_I00:NK.83">lt</a> <font color="Maroon" title="c8">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E8:104"><span class="lab"><font color="Green" title="E58">A5</font></span></a>, <a class="txt" href="#E3:104"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="txt" href="#E4:104"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="txt" href="#E9:104"><span class="lab"><font color="Green" title="E59">A6</font></span></a>, <a class="txt" href="#E6:104"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="txt" href="#E11:104"><span class="lab"><font color="Green" title="E61">A8</font></span></a>, <a class="txt" href="#E5:104"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="txt" href="#E10:104"><span class="lab"><font color="Green" title="E60">A7</font></span></a>, <a class="ref" href="aofa_i00.html#T24" target="_self" title="AOFA_I00:th.24">Th24</a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_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>
