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<span class="kw">let </span><font color="Maroon" title="c1">E</font>, <font color="Maroon" title="c2">D</font>, <font color="Maroon" title="c3">D9</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><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="Maroon" title="c2">D</font>,<font color="Maroon" title="c3">D9</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c1">E</font><br/>  for <font color="Olive" title="b4">T</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c2">D</font><br/>  for <font color="Olive" title="b5">T9</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c3">D9</font><br/>  for <font color="Olive" title="b6">S</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Olive" title="b2">j</font>,<font color="Maroon" title="c2">D</font><br/>  for <font color="Olive" title="b7">S9</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Olive" title="b2">j</font>,<font color="Maroon" title="c3">D9</font> holds  <font color="Olive" title="b3">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<span class="p1">(<span class="default"><font color="Olive" title="b4">T</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b6">S</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b5">T9</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b7">S9</font></span>)</span>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Olive" title="b4">T</font>,<font color="Olive" title="b5">T9</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Olive" title="b6">S</font>,<font color="Olive" title="b7">S9</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c5">j</font> be   <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="firebrick">::  thesis: </font></span></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="c2">D</font>,<font color="Maroon" title="c3">D9</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c1">E</font><br/>  for <font color="Olive" title="b2">T</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c2">D</font><br/>  for <font color="Olive" title="b3">T9</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c3">D9</font><br/>  for <font color="Olive" title="b4">S</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c2">D</font><br/>  for <font color="Olive" title="b5">S9</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c3">D9</font> holds  <font color="Olive" title="b1">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">T</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b4">S</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b3">T9</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b5">S9</font></span>)</span>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Olive" title="b2">T</font>,<font color="Olive" title="b3">T9</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Olive" title="b4">S</font>,<font color="Olive" title="b5">S9</font>)</span>)</span></span><br/><span class="kw">let </span><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="c2">D</font>,<font color="Maroon" title="c3">D9</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c1">E</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">T</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c2">D</font><br/>  for <font color="Olive" title="b2">T9</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c3">D9</font><br/>  for <font color="Olive" title="b3">S</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c2">D</font><br/>  for <font color="Olive" title="b4">S9</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c3">D9</font> holds  <font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">T</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b3">S</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">T9</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b4">S9</font></span>)</span>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Olive" title="b1">T</font>,<font color="Olive" title="b2">T9</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Olive" title="b3">S</font>,<font color="Olive" title="b4">S9</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">T</font> be   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c2">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">T9</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c3">D9</font><br/>  for <font color="Olive" title="b2">S</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c2">D</font><br/>  for <font color="Olive" title="b3">S9</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c3">D9</font> holds  <font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">S</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">T9</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b3">S9</font></span>)</span>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c7">T</font>,<font color="Olive" title="b1">T9</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Olive" title="b2">S</font>,<font color="Olive" title="b3">S9</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c8">T9</font> be   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c3">D9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c2">D</font><br/>  for <font color="Olive" title="b2">S9</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c3">D9</font> holds  <font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">S</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">T9</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">S9</font></span>)</span>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c7">T</font>,<font color="Maroon" title="c8">T9</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">S9</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c9">S</font> be   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c2">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S9</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c3">D9</font> holds  <font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">S</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">T9</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">S9</font></span>)</span>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c7">T</font>,<font color="Maroon" title="c8">T9</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c9">S</font>,<font color="Olive" title="b1">S9</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c10">S9</font> be   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c3">D9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">S</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">T9</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c10">S9</font></span>)</span>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c7">T</font>,<font color="Maroon" title="c8">T9</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c9">S</font>,<font color="Maroon" title="c10">S9</font>)</span>)</span></span><br/>



















<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">S</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of $1,<font color="Maroon" title="c2">D</font><br/>  for <font color="Olive" title="b2">S9</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of $1,<font color="Maroon" title="c3">D9</font> holds  <font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">S</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">T9</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">S9</font></span>)</span>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c7">T</font>,<font color="Maroon" title="c8">T9</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">S9</font>)</span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">j</font> be   <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="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">S</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c2">D</font><br/>  for <font color="Olive" title="b2">S9</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c3">D9</font> holds  <font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">S</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">T9</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">S9</font></span>)</span>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c7">T</font>,<font color="Maroon" title="c8">T9</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">S9</font>)</span>)</span> ) implies  for <font color="Olive" title="b1">S</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c11">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Maroon" title="c2">D</font><br/>  for <font color="Olive" title="b2">S9</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c11">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Maroon" title="c3">D9</font> holds  <font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">S</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">T9</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">S9</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c7">T</font>,<font color="Maroon" title="c8">T9</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">S9</font>)</span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:19_1"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
 for <font color="Olive" title="b1">S</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c2">D</font><br/>  for <font color="Olive" title="b2">S9</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c3">D9</font> holds  <font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">S</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">T9</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">S9</font></span>)</span>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c7">T</font>,<font color="Maroon" title="c8">T9</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">S9</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c11">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Maroon" title="c2">D</font><br/>  for <font color="Olive" title="b2">S9</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c11">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Maroon" title="c3">D9</font> holds  <font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">S</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">T9</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">S9</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c7">T</font>,<font color="Maroon" title="c8">T9</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">S9</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c12">S</font> be   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c11">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Maroon" title="c2">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S9</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c11">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Maroon" title="c3">D9</font> holds  <font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c12">S</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">T9</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">S9</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c7">T</font>,<font color="Maroon" title="c8">T9</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c12">S</font>,<font color="Olive" title="b1">S9</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c13">S9</font> be   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c11">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Maroon" title="c3">D9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c12">S</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">T9</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c13">S9</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c7">T</font>,<font color="Maroon" title="c8">T9</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c12">S</font>,<font color="Maroon" title="c13">S9</font>)</span>)</span></span><br/><span class="kw">consider </span><font color="Maroon" title="c14">S1</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c11">j</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c2">D</font>, <font color="Maroon" title="c15">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">D</font><span class="kw"> such that </span><br/><a NAME="E3:19_1"/><span class="lab"><font color="Green" title="E11">A2</font></span>: 
<font color="Maroon" title="c12">S</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c14">S1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c15">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T117" title="FINSEQ_2:th.117">FINSEQ_2:117</a></span>;<br/><a NAME="E4:19_1"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
<font color="Maroon" title="c7">T</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c12">S</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">S1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c15">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19_1"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="finseq_1.html#T32" title="FINSEQ_1:th.32">FINSEQ_1:32</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">S1</font> = <font color="Maroon" title="c14">S1</font> as   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c2">D</font> ;<br/><span class="kw">consider </span><font color="Maroon" title="c17">S19</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c11">j</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c3">D9</font>, <font color="Maroon" title="c18">d9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">D9</font><span class="kw"> such that </span><br/><a NAME="E7:19_1"/><span class="lab"><font color="Green" title="E13">A4</font></span>: 
<font color="Maroon" title="c13">S9</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">S19</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c18">d9</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T117" title="FINSEQ_2:th.117">FINSEQ_2:117</a></span>;<br/><a NAME="E8:19_1"/><span class="lab"><font color="Green" title="E14">A5</font></span>: 
<font color="Maroon" title="c8">T9</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c13">S9</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">T9</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c17">S19</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c18">d9</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:19_1"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="ref" href="finseq_1.html#T32" title="FINSEQ_1:th.32">FINSEQ_1:32</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">S19</font> = <font color="Maroon" title="c17">S19</font> as   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c3">D9</font> ;<br/><a NAME="E10:19_1"/>
( <font color="Maroon" title="c7">T</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">S1</font> is   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c8">T9</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c19">S19</font> is   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c3">D9</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T107" title="FINSEQ_2:th.107">FINSEQ_2:107</a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c12">S</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">T9</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c13">S9</font></span>)</span>) = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">S1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">T9</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c19">S19</font></span>)</span>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<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="c15">d</font>,<font color="Maroon" title="c18">d9</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19_1"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="txt" href="#E8:19_1"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="ref" href="finseqop.html#T10" target="_self" title="FINSEQOP:th.10">Th11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c7">T</font>,<font color="Maroon" title="c8">T9</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c16">S1</font>,<font color="Maroon" title="c19">S19</font>)</span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<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="c15">d</font>,<font color="Maroon" title="c18">d9</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1"><span class="lab"><font color="Green" title="E10">A1</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c7">T</font>,<font color="Maroon" title="c8">T9</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c16">S1</font>,<font color="Maroon" title="c19">S19</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<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="c15">d</font>,<font color="Maroon" title="c18">d9</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T32" title="FINSEQ_1:th.32">FINSEQ_1:32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c7">T</font>,<font color="Maroon" title="c8">T9</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c12">S</font>,<font color="Maroon" title="c13">S9</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19_1"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E7:19_1"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="ref" href="finseqop.html#T10" target="_self" title="FINSEQOP:th.10">Th11</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E2:19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A6</font></span>: 
 for <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">S</font> be   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c2">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S9</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c3">D9</font> holds  <font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">S</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">T9</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">S9</font></span>)</span>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c7">T</font>,<font color="Maroon" title="c8">T9</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c11">S</font>,<font color="Olive" title="b1">S9</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c12">S9</font> be   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c3">D9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">S</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">T9</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c12">S9</font></span>)</span>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c7">T</font>,<font color="Maroon" title="c8">T9</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c11">S</font>,<font color="Maroon" title="c12">S9</font>)</span>)</span></span><br/><a NAME="E1:19_2"/>
<font color="Maroon" title="c11">S</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c2">D</font>
 ;<br/><a NAME="E2:19_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A7</font></span>: 
<font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c11">S</font>,<font color="Maroon" title="c12">S9</font>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T73" title="FINSEQ_2:th.73">FINSEQ_2:73</a></span>;<br/><a NAME="E3:19_2"/>
( <font color="Maroon" title="c7">T</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">S</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c7">T</font> &amp; <font color="Maroon" title="c8">T9</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c12">S9</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c8">T9</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T95" title="FINSEQ_2:th.95">FINSEQ_2:95</a></span>;<br/><span class="kw">hence </span><a NAME="E4:19_2"/>
<font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">S</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">T9</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c12">S9</font></span>)</span>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c7">T</font>,<font color="Maroon" title="c8">T9</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c11">S</font>,<font color="Maroon" title="c12">S9</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_2"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="ref" href="finseq_1.html#T34" title="FINSEQ_1:th.34">FINSEQ_1:34</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A8</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E5:19"/>
 for <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E4:19"><span class="lab"><font color="Green" title="E11">A8</font></span></a>, <a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E10">A6</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E6:19"/>
<font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">S</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">T9</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c10">S9</font></span>)</span>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c7">T</font>,<font color="Maroon" title="c8">T9</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c9">S</font>,<font color="Maroon" title="c10">S9</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
