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<span class="kw">let </span><font color="Maroon" title="c1">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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">d</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">G</font> being   <a href="chain_1.html#M1" title="CHAIN_1:mode.1">Grating</a> of <font color="Olive" title="b1">d</font><br/> for <font color="Olive" title="b3">C1</font>, <font color="Olive" title="b4">C2</font> being  <a href="chain_1.html#NM4" title="CHAIN_1:NM.4">Chain</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b2">G</font> holds   <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">C1</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Olive" title="b4">C2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <font color="Olive" title="b3">C1</font></span>)</span> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <font color="Olive" title="b4">C2</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">d</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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">G</font> being   <a href="chain_1.html#M1" title="CHAIN_1:mode.1">Grating</a> of <font color="Maroon" title="c2">d</font><br/> for <font color="Olive" title="b2">C1</font>, <font color="Olive" title="b3">C2</font> being  <a href="chain_1.html#NM4" title="CHAIN_1:NM.4">Chain</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b1">G</font> holds   <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">C1</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Olive" title="b3">C2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <font color="Olive" title="b2">C1</font></span>)</span> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <font color="Olive" title="b3">C2</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">G</font> be    <a href="chain_1.html#M1" title="CHAIN_1:mode.1">Grating</a> of <font color="Maroon" title="c2">d</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">C1</font>, <font color="Olive" title="b2">C2</font> being  <a href="chain_1.html#NM4" title="CHAIN_1:NM.4">Chain</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c3">G</font> holds   <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">C1</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Olive" title="b2">C2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <font color="Olive" title="b1">C1</font></span>)</span> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <font color="Olive" title="b2">C2</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">C1</font>, <font color="Maroon" title="c5">C2</font> be   <a href="chain_1.html#NM4" title="CHAIN_1:NM.4">Chain</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c3">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C1</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Maroon" title="c5">C2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <font color="Maroon" title="c4">C1</font></span>)</span> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <font color="Maroon" title="c5">C2</font></span>)</span></span><br/>







<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">A</font> be   <a href="chain_1.html#NM3" title="CHAIN_1:NM.3">Cell</a> of <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">G</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="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C1</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Maroon" title="c5">C2</font></span>)</span> iff <font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <font color="Maroon" title="c4">C1</font></span>)</span> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <font color="Maroon" title="c5">C2</font></span>)</span> )</span><br/><a NAME="E1:86_1"/><span class="lab"><font color="Green" title="E59">A1</font></span>: 
<span class="p1">(<span class="default"><a href="chain_1.html#K9" title="CHAIN_1:func.9">star</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C1</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c5">C2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="chain_1.html#K9" title="CHAIN_1:func.9">star</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">C1</font></span>)</span> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="chain_1.html#K9" title="CHAIN_1:func.9">star</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">C2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T112" title="XBOOLE_1:th.112">XBOOLE_1:112</a></span>;<br/><a NAME="E2:86_1"/><span class="lab"><font color="Green" title="E60">A2</font></span>: 
( <font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C1</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Maroon" title="c5">C2</font></span>)</span> iff ( <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">d</font> &amp; not  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="chain_1.html#K9" title="CHAIN_1:func.9">star</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">C1</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c5">C2</font></span>)</span></span>)</span> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chain_1.html#T52" target="_self" title="CHAIN_1:th.52">Th52</a></span>;<br/><a NAME="E3:86_1"/><span class="lab"><font color="Green" title="E61">A3</font></span>: 
( <font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <font color="Maroon" title="c4">C1</font> iff ( <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">d</font> &amp; not  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="chain_1.html#K9" title="CHAIN_1:func.9">star</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">C1</font></span>)</span> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chain_1.html#T52" target="_self" title="CHAIN_1:th.52">Th52</a></span>;<br/><a NAME="E4:86_1"/>
( <font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <font color="Maroon" title="c5">C2</font> iff ( <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">d</font> &amp; not  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="chain_1.html#K9" title="CHAIN_1:func.9">star</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">C2</font></span>)</span> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chain_1.html#T52" target="_self" title="CHAIN_1:th.52">Th52</a></span>;<br/><span class="kw">hence </span><a NAME="E5:86_1"/>
( <font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C1</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Maroon" title="c5">C2</font></span>)</span> iff <font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <font color="Maroon" title="c4">C1</font></span>)</span> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <font color="Maroon" title="c5">C2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_1"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="txt" href="#E2:86_1"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="txt" href="#E3:86_1"><span class="lab"><font color="Green" title="E61">A3</font></span></a>, <a class="ref" href="chain_1.html#T9" target="_self" title="CHAIN_1:th.9">Th9</a>, <a class="ref" href="xboole_0.html#T1" title="XBOOLE_0:th.1">XBOOLE_0:1</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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:86"/>
 <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C1</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Maroon" title="c5">C2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <font color="Maroon" title="c4">C1</font></span>)</span> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <font color="Maroon" title="c5">C2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T8" title="SUBSET_1:th.8">SUBSET_1:8</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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