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<span class="kw">let </span><font color="Maroon" title="c2">D1</font>, <font color="Maroon" title="c3">D2</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> </span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> </span>)</span></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span>; <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="c2">D1</font> <a href="cqc_sim1.html#R1" title="CQC_SIM1:pred.1">is_Sep-closed_on</a> <font color="Maroon" title="c1">p</font> &amp; ( for <font color="Olive" title="b1">D</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> </span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> </span>)</span></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span>  st <font color="Olive" title="b1">D</font> <a href="cqc_sim1.html#R1" title="CQC_SIM1:pred.1">is_Sep-closed_on</a> <font color="Maroon" title="c1">p</font> holds <br/><font color="Maroon" title="c2">D1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">D</font> ) &amp; <font color="Maroon" title="c3">D2</font> <a href="cqc_sim1.html#R1" title="CQC_SIM1:pred.1">is_Sep-closed_on</a> <font color="Maroon" title="c1">p</font> &amp; ( for <font color="Olive" title="b1">D</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> </span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> </span>)</span></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span>  st <font color="Olive" title="b1">D</font> <a href="cqc_sim1.html#R1" title="CQC_SIM1:pred.1">is_Sep-closed_on</a> <font color="Maroon" title="c1">p</font> holds <br/><font color="Maroon" title="c3">D2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">D</font> ) implies <font color="Maroon" title="c2">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:52_1_2"/><span class="lab"><font color="Green" title="E27">A17</font></span>: 
<font color="Maroon" title="c2">D1</font> <a href="cqc_sim1.html#R1" title="CQC_SIM1:pred.1">is_Sep-closed_on</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">and </span><br/><a NAME="E2:52_1_2"/><span class="lab"><font color="Green" title="E28">A18</font></span>: 
for <font color="Olive" title="b1">D</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> </span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> </span>)</span></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span>  st <font color="Olive" title="b1">D</font> <a href="cqc_sim1.html#R1" title="CQC_SIM1:pred.1">is_Sep-closed_on</a> <font color="Maroon" title="c1">p</font> holds <br/><font color="Maroon" title="c2">D1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">D</font>
 <span class="kw">and </span><br/><a NAME="E3:52_1_2"/><span class="lab"><font color="Green" title="E29">A19</font></span>: 
<font color="Maroon" title="c3">D2</font> <a href="cqc_sim1.html#R1" title="CQC_SIM1:pred.1">is_Sep-closed_on</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">and </span><br/><a NAME="E4:52_1_2"/><span class="lab"><font color="Green" title="E30">A20</font></span>: 
for <font color="Olive" title="b1">D</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> </span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> </span>)</span></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span>  st <font color="Olive" title="b1">D</font> <a href="cqc_sim1.html#R1" title="CQC_SIM1:pred.1">is_Sep-closed_on</a> <font color="Maroon" title="c1">p</font> holds <br/><font color="Maroon" title="c3">D2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">D</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="c2">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D2</font></span><br/>

<span class="kw">thus </span><a NAME="E5:52_1_2"/>
( <font color="Maroon" title="c2">D1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">D2</font> &amp; <font color="Maroon" title="c3">D2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">D1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_2"><span class="lab"><font color="Green" title="E27">A17</font></span></a>, <a class="txt" href="#E2:52_1_2"><span class="lab"><font color="Green" title="E28">A18</font></span></a>, <a class="txt" href="#E3:52_1_2"><span class="lab"><font color="Green" title="E29">A19</font></span></a>, <a class="txt" href="#E4:52_1_2"><span class="lab"><font color="Green" title="E30">A20</font></span></a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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