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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E39">Th27</font></span>: <a NAME="T28"><span class="comment"><font color="firebrick">:: AOFA_L00:28</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">number</a> <br/>  for <font color="Olive" title="b2">J</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   non  <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>  <a href="catalg_1.html#NM1" title="CATALG_1:NM.1">Signature</a><br/>  for <font color="Olive" title="b3">T</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Olive" title="b2">J</font><br/>  for <font color="Olive" title="b4">X</font> being  <a href="relat_1.html#V3" title="RELAT_1:attr.3">V3</a>()  <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Olive" title="b3">T</font><br/>  for <font color="Olive" title="b5">S</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   non  <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>  <font color="Olive">b<sub>2</sub></font> <a href="aofa_l00.html#V2" title="AOFA_L00:attr.2">-extension</a>  <font color="Olive">b<sub>1</sub></font> <a href="aofa_l00.html#V6" title="AOFA_L00:attr.6">PC-correct</a>   <a href="aofa_l00.html#V7" title="AOFA_L00:attr.7">QC-correct</a>   <a href="aofa_l00.html#L2" title="AOFA_L00:struct.2">QCLangSignature</a> over  <a href="aofa_l00.html#K4" title="AOFA_L00:func.4">Union</a> <font color="Olive" title="b4">X</font><br/>  for <font color="Olive" title="b6">Y</font> being  <a href="relat_1.html#V3" title="RELAT_1:attr.3">V3</a>() <font color="Olive">b<sub>4</sub></font> <a href="aofa_l00.html#V1" title="AOFA_L00:attr.1">-tolerating</a>  <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b5">S</font><br/>  for <font color="Olive" title="b7">L</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="aofa_l00.html#NM2" title="AOFA_L00:NM.2">Language</a> of <font color="Olive" title="b6">Y</font>,<font color="Olive" title="b5">S</font>  st <font color="Olive" title="b7">L</font> is  <a href="aofa_l00.html#V12" title="AOFA_L00:attr.12">subst-correct</a>  &amp; <font color="Olive" title="b6">Y</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</a> of  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Olive" title="b7">L</font> holds <br/> for <font color="Olive" title="b8">x</font>, <font color="Olive" title="b9">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Olive" title="b6">Y</font><br/>  for <font color="Olive" title="b10">a</font> being   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Olive" title="b5">S</font>  st <font color="Olive" title="b8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b6">Y</font> <a href="msafree4.html#K1" title="MSAFREE4:func.1">.</a> <font color="Olive" title="b10">a</font> &amp; <font color="Olive" title="b9">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b6">Y</font> <a href="msafree4.html#K1" title="MSAFREE4:func.1">.</a> <font color="Olive" title="b10">a</font> holds <br/> for <font color="Olive" title="b11">A</font> being   <a href="aofa_l00.html#NM3" title="AOFA_L00:NM.3">Formula</a> of <font color="Olive" title="b7">L</font> holds <br/> ( <span class="p1">(<span class="default"><a href="aofa_l00.html#K12" title="AOFA_L00:func.12">\not</a> <font color="Olive" title="b11">A</font></span>)</span> <a href="aofa_l00.html#K6" title="AOFA_L00:func.6">/</a> (<font color="Olive" title="b8">x</font>,<font color="Olive" title="b9">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aofa_l00.html#K12" title="AOFA_L00:func.12">\not</a> <span class="p1">(<span class="default"><font color="Olive" title="b11">A</font> <a href="aofa_l00.html#K6" title="AOFA_L00:func.6">/</a> (<font color="Olive" title="b8">x</font>,<font color="Olive" title="b9">y</font>)</span>)</span> &amp; (  for <font color="Olive" title="b12">B</font> being   <a href="aofa_l00.html#NM3" title="AOFA_L00:NM.3">Formula</a> of <font color="Olive" title="b7">L</font> holds <br/> ( <span class="p1">(<span class="default"><font color="Olive" title="b11">A</font> <a href="aofa_l00.html#K13" title="AOFA_L00:func.13">\and</a> <font color="Olive" title="b12">B</font></span>)</span> <a href="aofa_l00.html#K6" title="AOFA_L00:func.6">/</a> (<font color="Olive" title="b8">x</font>,<font color="Olive" title="b9">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b11">A</font> <a href="aofa_l00.html#K6" title="AOFA_L00:func.6">/</a> (<font color="Olive" title="b8">x</font>,<font color="Olive" title="b9">y</font>)</span>)</span> <a href="aofa_l00.html#K13" title="AOFA_L00:func.13">\and</a> <span class="p1">(<span class="default"><font color="Olive" title="b12">B</font> <a href="aofa_l00.html#K6" title="AOFA_L00:func.6">/</a> (<font color="Olive" title="b8">x</font>,<font color="Olive" title="b9">y</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b11">A</font> <a href="aofa_l00.html#K14" title="AOFA_L00:func.14">\or</a> <font color="Olive" title="b12">B</font></span>)</span> <a href="aofa_l00.html#K6" title="AOFA_L00:func.6">/</a> (<font color="Olive" title="b8">x</font>,<font color="Olive" title="b9">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b11">A</font> <a href="aofa_l00.html#K6" title="AOFA_L00:func.6">/</a> (<font color="Olive" title="b8">x</font>,<font color="Olive" title="b9">y</font>)</span>)</span> <a href="aofa_l00.html#K14" title="AOFA_L00:func.14">\or</a> <span class="p1">(<span class="default"><font color="Olive" title="b12">B</font> <a href="aofa_l00.html#K6" title="AOFA_L00:func.6">/</a> (<font color="Olive" title="b8">x</font>,<font color="Olive" title="b9">y</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b11">A</font> <a href="aofa_l00.html#K15" title="AOFA_L00:func.15">\imp</a> <font color="Olive" title="b12">B</font></span>)</span> <a href="aofa_l00.html#K6" title="AOFA_L00:func.6">/</a> (<font color="Olive" title="b8">x</font>,<font color="Olive" title="b9">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b11">A</font> <a href="aofa_l00.html#K6" title="AOFA_L00:func.6">/</a> (<font color="Olive" title="b8">x</font>,<font color="Olive" title="b9">y</font>)</span>)</span> <a href="aofa_l00.html#K15" title="AOFA_L00:func.15">\imp</a> <span class="p1">(<span class="default"><font color="Olive" title="b12">B</font> <a href="aofa_l00.html#K6" title="AOFA_L00:func.6">/</a> (<font color="Olive" title="b8">x</font>,<font color="Olive" title="b9">y</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b11">A</font> <a href="aofa_l00.html#K16" title="AOFA_L00:func.16">\iff</a> <font color="Olive" title="b12">B</font></span>)</span> <a href="aofa_l00.html#K6" title="AOFA_L00:func.6">/</a> (<font color="Olive" title="b8">x</font>,<font color="Olive" title="b9">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b11">A</font> <a href="aofa_l00.html#K6" title="AOFA_L00:func.6">/</a> (<font color="Olive" title="b8">x</font>,<font color="Olive" title="b9">y</font>)</span>)</span> <a href="aofa_l00.html#K16" title="AOFA_L00:func.16">\iff</a> <span class="p1">(<span class="default"><font color="Olive" title="b12">B</font> <a href="aofa_l00.html#K6" title="AOFA_L00:func.6">/</a> (<font color="Olive" title="b8">x</font>,<font color="Olive" title="b9">y</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="aofa_l00.html#K11" title="AOFA_L00:func.11">\true_</a> <font color="Olive" title="b7">L</font></span>)</span> <a href="aofa_l00.html#K6" title="AOFA_L00:func.6">/</a> (<font color="Olive" title="b8">x</font>,<font color="Olive" title="b9">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aofa_l00.html#K11" title="AOFA_L00:func.11">\true_</a> <font color="Olive" title="b7">L</font> ) ) )</div></div>
