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<b>let </b><font color="Maroon" title="c1">A</font> be   <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">C</font>, <font color="Olive" title="b2">I</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font><br/> for <font color="Olive" title="b3">S</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b4">T</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Olive" title="b3">S</font><br/> for <font color="Olive" title="b5">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b3">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Olive" title="b3">S</font>  st <font color="Olive" title="b5">f</font> is <a href="aofa_000.html#V13" title="AOFA_000:attr.13">complying_with_empty-instruction</a> &amp; <font color="Olive" title="b5">f</font> <a href="aofa_000.html#R2" title="AOFA_000:pred.2">complies_with_if_wrt</a> <font color="Olive" title="b4">T</font> holds <br/>for <font color="Olive" title="b6">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b3">S</font>  st <font color="Olive" title="b5">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b6">s</font>,<font color="Olive" title="b1">C</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Olive" title="b4">T</font> holds <br/><font color="Olive" title="b5">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b6">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <font color="Olive" title="b1">C</font>,<font color="Olive" title="b2">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b6">s</font>,<font color="Olive" title="b1">C</font></span><br/><b>let </b><font color="Maroon" title="c2">C</font>, <font color="Maroon" title="c3">I</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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 <font color="Olive" title="b1">S</font><br/> for <font color="Olive" title="b3">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Olive" title="b1">S</font>  st <font color="Olive" title="b3">f</font> is <a href="aofa_000.html#V13" title="AOFA_000:attr.13">complying_with_empty-instruction</a> &amp; <font color="Olive" title="b3">f</font> <a href="aofa_000.html#R2" title="AOFA_000:pred.2">complies_with_if_wrt</a> <font color="Olive" title="b2">T</font> holds <br/>for <font color="Olive" title="b4">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">S</font>  st <font color="Olive" title="b3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b4">s</font>,<font color="Maroon" title="c2">C</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Olive" title="b2">T</font> holds <br/><font color="Olive" title="b3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b4">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b4">s</font>,<font color="Maroon" title="c2">C</font></span><br/><b>let </b><font color="Maroon" title="c4">S</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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">T</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c4">S</font><br/> for <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">S</font>  st <font color="Olive" title="b2">f</font> is <a href="aofa_000.html#V13" title="AOFA_000:attr.13">complying_with_empty-instruction</a> &amp; <font color="Olive" title="b2">f</font> <a href="aofa_000.html#R2" title="AOFA_000:pred.2">complies_with_if_wrt</a> <font color="Olive" title="b1">T</font> holds <br/>for <font color="Olive" title="b3">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">S</font>  st <font color="Olive" title="b2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b3">s</font>,<font color="Maroon" title="c2">C</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Olive" title="b1">T</font> holds <br/><font color="Olive" title="b2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b3">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b3">s</font>,<font color="Maroon" title="c2">C</font></span><br/><b>let </b><font color="Maroon" title="c5">T</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c4">S</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">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">S</font>  st <font color="Olive" title="b1">f</font> is <a href="aofa_000.html#V13" title="AOFA_000:attr.13">complying_with_empty-instruction</a> &amp; <font color="Olive" title="b1">f</font> <a href="aofa_000.html#R2" title="AOFA_000:pred.2">complies_with_if_wrt</a> <font color="Maroon" title="c5">T</font> holds <br/>for <font color="Olive" title="b2">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">S</font>  st <font color="Olive" title="b1">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b2">s</font>,<font color="Maroon" title="c2">C</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c5">T</font> holds <br/><font color="Olive" title="b1">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b2">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b2">s</font>,<font color="Maroon" title="c2">C</font></span><br/><b>let </b><font color="Maroon" title="c6">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">f</font> is <a href="aofa_000.html#V13" title="AOFA_000:attr.13">complying_with_empty-instruction</a> &amp; <font color="Maroon" title="c6">f</font> <a href="aofa_000.html#R2" title="AOFA_000:pred.2">complies_with_if_wrt</a> <font color="Maroon" title="c5">T</font> implies for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">S</font>  st <font color="Maroon" title="c6">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="c2">C</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c5">T</font> holds <br/><font color="Maroon" title="c6">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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="c2">C</font> )</span><br/>











<b>assume </b><b>that </b><br/><a NAME="E1:132"/><i><font color="Green" title="E85">A1</font></i>: 
<font color="Maroon" title="c6">f</font> is <a href="aofa_000.html#V13" title="AOFA_000:attr.13">complying_with_empty-instruction</a>
 <b>and </b><br/><a NAME="E2:132"/><i><font color="Green" title="E86">A2</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="aofa_000.html#R2" title="AOFA_000:pred.2">complies_with_if_wrt</a> <font color="Maroon" title="c5">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">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">S</font>  st <font color="Maroon" title="c6">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="c2">C</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c5">T</font> holds <br/><font color="Maroon" title="c6">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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="c2">C</font></span><br/>

<b>let </b><font color="Maroon" title="c7">s</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c2">C</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c5">T</font> implies <font color="Maroon" title="c6">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c2">C</font> )</span><br/>

<b>assume </b><a NAME="E3:132"/>
<font color="Maroon" title="c6">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c2">C</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c5">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c2">C</font></span><br/>

<b>hence </b><font color="Maroon" title="c6">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">I</font></span>)</span> = 
<font color="Maroon" title="c6">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c2">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="aofa_000.html#K12" title="AOFA_000:func.12">EmptyIns</a> <font color="Maroon" title="c1">A</font></span>)</span>
<b>by </b><i><a class="ref" href="aofa_000.html#D30" target="_self" title="AOFA_000:def.30">Def30</a>, <a class="txt" href="#E2:132"><i><font color="Green" title="E86">A2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c2">C</font>
<b>by </b><i><a class="ref" href="aofa_000.html#D28" target="_self" title="AOFA_000:def.28">Def28</a>, <a class="txt" href="#E1:132"><i><font color="Green" title="E85">A1</font></i></a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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