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<span class="kw">let </span><font color="Maroon" title="c1">D1</font>, <font color="Maroon" title="c2">D2</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">b1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D1</font><br/> for <font color="Olive" title="b3">a2</font>, <font color="Olive" title="b4">b2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">D2</font><br/> for <font color="Olive" title="b5">f1</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D1</font><br/> for <font color="Olive" title="b6">f2</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">D2</font> holds  <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Olive" title="b5">f1</font>,<font color="Olive" title="b6">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">a1</font>,<font color="Olive" title="b3">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">b1</font>,<font color="Olive" title="b4">b2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">f1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">a1</font>,<font color="Olive" title="b2">b1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b6">f2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b3">a2</font>,<font color="Olive" title="b4">b2</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">a1</font>, <font color="Maroon" title="c4">b1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D1</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">a2</font>, <font color="Olive" title="b2">b2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">D2</font><br/> for <font color="Olive" title="b3">f1</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D1</font><br/> for <font color="Olive" title="b4">f2</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">D2</font> holds  <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Olive" title="b3">f1</font>,<font color="Olive" title="b4">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Olive" title="b1">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">b1</font>,<font color="Olive" title="b2">b2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">f1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">f2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">a2</font>,<font color="Olive" title="b2">b2</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">a2</font>, <font color="Maroon" title="c6">b2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">D2</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">f1</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D1</font><br/> for <font color="Olive" title="b2">f2</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">D2</font> holds  <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c5">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c6">b2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">f2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">f1</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D1</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">f2</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">D2</font> holds  <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c7">f1</font>,<font color="Olive" title="b1">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c5">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c6">b2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">f2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c8">f2</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">D2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c7">f1</font>,<font color="Maroon" title="c8">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c5">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c6">b2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">f2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><br/>















<a NAME="E1:33"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c7">f1</font>,<font color="Maroon" title="c8">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">D1</font>,<font color="Maroon" title="c2">D2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">D1</font>,<font color="Maroon" title="c2">D2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E2:33"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c5">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c6">b2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">D1</font>,<font color="Maroon" title="c2">D2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">D1</font>,<font color="Maroon" title="c2">D2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
;<br/>
<span class="kw">hence </span><a NAME="E3:33"/>
<span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c7">f1</font>,<font color="Maroon" title="c8">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c5">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c6">b2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">f2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="funct_4.html#T58" title="FUNCT_4:th.58">FUNCT_4:58</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/>


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