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<b>let </b><font color="Maroon" title="c1">A</font> be  <a href="aofa_i00.html#V2" title="AOFA_I00:attr.2">Euclidean</a> <a href="aofa_000.html#NM1" title="AOFA_000:NM.1">preIfWhileAlgebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="card_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">T</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <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><br/> for <font color="Olive" title="b4">f</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> ,<font color="Olive" title="b3">T</font><br/> for <font color="Olive" title="b5">v</font> being   <a href="aofa_i00.html#M2" title="AOFA_I00:mode.2">INT-Variable</a> of <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b4">f</font><br/> for <font color="Olive" title="b6">t</font> being   <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b4">f</font> holds <br/> ( <span class="p1">(<span class="default"><font color="Olive" title="b4">f</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">v</font> <a href="aofa_i00.html#K44" title="AOFA_I00:func.44">:=</a> <font color="Olive" title="b6">t</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">s</font> &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="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">s</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b4">f</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">v</font> <a href="aofa_i00.html#K44" title="AOFA_I00:func.44">:=</a> <font color="Olive" title="b6">t</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">T</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <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><br/> for <font color="Olive" title="b3">f</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> ,<font color="Olive" title="b2">T</font><br/> for <font color="Olive" title="b4">v</font> being   <a href="aofa_i00.html#M2" title="AOFA_I00:mode.2">INT-Variable</a> of <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b3">f</font><br/> for <font color="Olive" title="b5">t</font> being   <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b3">f</font> holds <br/> ( <span class="p1">(<span class="default"><font color="Olive" title="b3">f</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">v</font> <a href="aofa_i00.html#K44" title="AOFA_I00:func.44">:=</a> <font color="Olive" title="b5">t</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> &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="b4">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b3">f</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">v</font> <a href="aofa_i00.html#K44" title="AOFA_I00:func.44">:=</a> <font color="Olive" title="b5">t</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">T</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <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><br/> for <font color="Olive" title="b2">f</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> ,<font color="Olive" title="b1">T</font><br/> for <font color="Olive" title="b3">v</font> being   <a href="aofa_i00.html#M2" title="AOFA_I00:mode.2">INT-Variable</a> of <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b2">f</font><br/> for <font color="Olive" title="b4">t</font> being   <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b2">f</font> holds <br/> ( <span class="p1">(<span class="default"><font color="Olive" title="b2">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="b3">v</font> <a href="aofa_i00.html#K44" title="AOFA_I00:func.44">:=</a> <font color="Olive" title="b4">t</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font> &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="b3">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b2">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="b3">v</font> <a href="aofa_i00.html#K44" title="AOFA_I00:func.44">:=</a> <font color="Olive" title="b4">t</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">T</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <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 class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</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> ,<font color="Maroon" title="c4">T</font><br/> for <font color="Olive" title="b2">v</font> being   <a href="aofa_i00.html#M2" title="AOFA_I00:mode.2">INT-Variable</a> of <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">f</font><br/> for <font color="Olive" title="b3">t</font> being   <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">f</font> holds <br/> ( <span class="p1">(<span class="default"><font color="Olive" title="b1">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="b2">v</font> <a href="aofa_i00.html#K44" title="AOFA_I00:func.44">:=</a> <font color="Olive" title="b3">t</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font> &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="Olive" title="b2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">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="b2">v</font> <a href="aofa_i00.html#K44" title="AOFA_I00:func.44">:=</a> <font color="Olive" title="b3">t</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> ,<font color="Maroon" title="c4">T</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">v</font> being   <a href="aofa_i00.html#M2" title="AOFA_I00:mode.2">INT-Variable</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c5">f</font><br/> for <font color="Olive" title="b2">t</font> being   <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c5">f</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">v</font> <a href="aofa_i00.html#K44" title="AOFA_I00:func.44">:=</a> <font color="Olive" title="b2">t</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font> &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="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</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">v</font> <a href="aofa_i00.html#K44" title="AOFA_I00:func.44">:=</a> <font color="Olive" title="b2">t</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">v</font> be    <a href="aofa_i00.html#M2" title="AOFA_I00:mode.2">INT-Variable</a> of <font color="Maroon" title="c1">A</font>,<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"> for <font color="Olive" title="b1">t</font> being   <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c5">f</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">v</font> <a href="aofa_i00.html#K44" title="AOFA_I00:func.44">:=</a> <font color="Olive" title="b1">t</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font> &amp; ( for <font color="Olive" title="b2">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="b2">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</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">v</font> <a href="aofa_i00.html#K44" title="AOFA_I00:func.44">:=</a> <font color="Olive" title="b1">t</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b2">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="b2">z</font> ) )</span><br/><b>let </b><font color="Maroon" title="c7">t</font> be    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of <font color="Maroon" title="c1">A</font>,<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"> ( <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">v</font> <a href="aofa_i00.html#K44" title="AOFA_I00:func.44">:=</a> <font color="Maroon" title="c7">t</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font> &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="c6">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</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">v</font> <a href="aofa_i00.html#K44" title="AOFA_I00:func.44">:=</a> <font color="Maroon" title="c7">t</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/>













<a NAME="E1:89"/>
<font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c7">t</font> <a href="aofa_i00.html#R2" title="AOFA_I00:pred.2">form_assignment_wrt</a> <font color="Maroon" title="c5">f</font>
 
<b>by </b><i><a class="ref" href="aofa_i00.html#D22" target="_self" title="AOFA_I00:def.22">INT'iwa</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">I0</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E3:89"/><i><font color="Green" title="E42">A0</font></i>: 
<font color="Maroon" title="c8">I0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon" title="c1">A</font>
 <b>and </b><br/><a NAME="E4:89"/><i><font color="Green" title="E43">A1</font></i>: 
for <font color="Olive" title="b1">s</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  holds  <font color="Maroon" title="c5">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,<font color="Maroon" title="c8">I0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font></span>)</span>
 <b>by </b><i><a class="ref" href="aofa_i00.html#D15" target="_self" title="AOFA_I00:def.15">FA</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">Y</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">I</font> where <font color="Olive" title="b1">I</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font> : ( <font color="Olive" title="b1">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon" title="c1">A</font> &amp; ( 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="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  holds  <font color="Maroon" title="c5">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b2">s</font>,<font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">s</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">s</font></span>)</span> ) ) </span>}</span> ;<br/>
<a NAME="E5:89"/>
<font color="Maroon" title="c8">I0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">I</font> where <font color="Olive" title="b1">I</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font> : ( <font color="Olive" title="b1">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon" title="c1">A</font> &amp; ( 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="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  holds  <font color="Maroon" title="c5">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b2">s</font>,<font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">s</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">s</font></span>)</span> ) ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E3:89"><i><font color="Green" title="E42">A0</font></i></a>, <a class="txt" href="aofa_i00.html#E4:89"><i><font color="Green" title="E43">A1</font></i></a></i>;<br/>
<a NAME="E6:89"/><b>then </b>
<font color="Maroon" title="c6">v</font> <a href="aofa_i00.html#K44" title="AOFA_I00:func.44">:=</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">I</font> where <font color="Olive" title="b1">I</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font> : ( <font color="Olive" title="b1">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon" title="c1">A</font> &amp; ( 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="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  holds  <font color="Maroon" title="c5">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b2">s</font>,<font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">s</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">s</font></span>)</span> ) ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon" title="c10">I</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E8:89"/><i><font color="Green" title="E44">A2</font></i>: 
<font color="Maroon" title="c6">v</font> <a href="aofa_i00.html#K44" title="AOFA_I00:func.44">:=</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">I</font>
 <b>and </b><br/><a NAME="E9:89"/>
<font color="Maroon" title="c10">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon" title="c1">A</font>
 <b>and </b><br/><a NAME="E10:89"/><i><font color="Green" title="E45">A3</font></i>: 
for <font color="Olive" title="b1">s</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  holds  <font color="Maroon" title="c5">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,<font color="Maroon" title="c10">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font></span>)</span>
 ;<br/>
<a NAME="E11:89"/><i><font color="Green" title="E46">A4</font></i>: 
<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="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="aofa_i00.html#K44" title="AOFA_I00:func.44">:=</a> <font color="Maroon" title="c7">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <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"><font color="Maroon" title="c7">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font></span>)</span>
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E8:89"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="aofa_i00.html#E10:89"><i><font color="Green" title="E45">A3</font></i></a></i>;<br/>
<a NAME="E12:89"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E13:89"/>
( <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">v</font> <a href="aofa_i00.html#K44" title="AOFA_I00:func.44">:=</a> <font color="Maroon" title="c7">t</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font> &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="c6">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</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">v</font> <a href="aofa_i00.html#K44" title="AOFA_I00:func.44">:=</a> <font color="Maroon" title="c7">t</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#E11:89"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</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>
