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<span class="kw">let </span><font color="Maroon" title="c1">m</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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">z1</font>, <font color="Olive" title="b2">z2</font> being  <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c1">m</font>, <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> <br/> for <font color="Olive" title="b3">d1</font>, <font color="Olive" title="b4">d2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">z1</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Olive" title="b3">d1</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="binari_2.html#K6" title="BINARI_2:func.6">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">z2</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Olive" title="b4">d2</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">z1</font> <a href="binarith.html#K7" title="BINARITH:func.7">+</a> <span class="p2">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Neg2</a> <font color="Olive" title="b2">z2</font></span>)</span></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b3">d1</font> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p5">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <font color="Olive" title="b4">d2</font></span>)</span></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p4">(<span class="default"><a href="binarith.html#K8" title="BINARITH:func.8">add_ovfl</a> <span class="p5">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Olive" title="b2">z2</font></span>)</span>,<span class="p5">(<span class="default"><a href="binari_2.html#K1" title="BINARI_2:func.1">Bin1</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p3">(<span class="default"><a href="binarith.html#K8" title="BINARITH:func.8">add_ovfl</a> <font color="Olive" title="b1">z1</font>,<span class="p4">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Neg2</a> <font color="Olive" title="b2">z2</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">z1</font>, <font color="Maroon" title="c3">z2</font> be   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c1">m</font>, <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</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">d1</font>, <font color="Olive" title="b2">d2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Olive" title="b1">d1</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="binari_2.html#K6" title="BINARI_2:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Olive" title="b2">d2</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K7" title="BINARITH:func.7">+</a> <span class="p2">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Neg2</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">d1</font> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p5">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <font color="Olive" title="b2">d2</font></span>)</span></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p4">(<span class="default"><a href="binarith.html#K8" title="BINARITH:func.8">add_ovfl</a> <span class="p5">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c3">z2</font></span>)</span>,<span class="p5">(<span class="default"><a href="binari_2.html#K1" title="BINARI_2:func.1">Bin1</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p3">(<span class="default"><a href="binarith.html#K8" title="BINARITH:func.8">add_ovfl</a> <font color="Maroon" title="c2">z1</font>,<span class="p4">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Neg2</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">d1</font>, <font color="Maroon" title="c5">d2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">d1</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="binari_2.html#K6" title="BINARI_2:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">d2</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K7" title="BINARITH:func.7">+</a> <span class="p2">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Neg2</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">d1</font> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p5">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <font color="Maroon" title="c5">d2</font></span>)</span></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p4">(<span class="default"><a href="binarith.html#K8" title="BINARITH:func.8">add_ovfl</a> <span class="p5">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c3">z2</font></span>)</span>,<span class="p5">(<span class="default"><a href="binari_2.html#K1" title="BINARI_2:func.1">Bin1</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p3">(<span class="default"><a href="binarith.html#K8" title="BINARITH:func.8">add_ovfl</a> <font color="Maroon" title="c2">z1</font>,<span class="p4">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Neg2</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span><br/>







<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">d1</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="binari_2.html#K6" title="BINARI_2:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">d2</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">d1</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="binarith.html#K7" title="BINARITH:func.7">+</a> <span class="p1">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Neg2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p3"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">d2</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binari_2.html#T19" target="_self" title="BINARI_2:th.19">Th19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">d1</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="binarith.html#K7" title="BINARITH:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Neg2</a> <font color="Maroon" title="c3">z2</font></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <font color="Maroon" title="c5">d2</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p4">(<span class="default"><a href="binarith.html#K8" title="BINARITH:func.8">add_ovfl</a> <span class="p5">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c3">z2</font></span>)</span>,<span class="p5">(<span class="default"><a href="binari_2.html#K1" title="BINARI_2:func.1">Bin1</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binari_2.html#T16" target="_self" title="BINARI_2:th.16">Th16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K7" title="BINARITH:func.7">+</a> <span class="p2">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Neg2</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">d1</font> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <font color="Maroon" title="c5">d2</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p5">(<span class="default"><a href="binarith.html#K8" title="BINARITH:func.8">add_ovfl</a> <span class="p0">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c3">z2</font></span>)</span>,<span class="p0">(<span class="default"><a href="binari_2.html#K1" title="BINARI_2:func.1">Bin1</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p3">(<span class="default"><a href="binarith.html#K8" title="BINARITH:func.8">add_ovfl</a> <font color="Maroon" title="c2">z1</font>,<span class="p4">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Neg2</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binarith.html#T45" title="BINARITH:th.45">BINARITH:45</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K7" title="BINARITH:func.7">+</a> <span class="p2">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Neg2</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">d1</font> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p5">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <font color="Maroon" title="c5">d2</font></span>)</span></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p4">(<span class="default"><a href="binarith.html#K8" title="BINARITH:func.8">add_ovfl</a> <span class="p5">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c3">z2</font></span>)</span>,<span class="p5">(<span class="default"><a href="binari_2.html#K1" title="BINARI_2:func.1">Bin1</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p3">(<span class="default"><a href="binarith.html#K8" title="BINARITH:func.8">add_ovfl</a> <font color="Maroon" title="c2">z1</font>,<span class="p4">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Neg2</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboolean.html#T73" title="XBOOLEAN:th.73">XBOOLEAN:73</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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