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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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="88_1"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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="E1:88_1"/><i><font color="Green" title="E61">A1</font></i>: 
<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>
 <b>by </b><i><a class="ref" href="xboole_1.html#T112" title="XBOOLE_1:th.112">XBOOLE_1:112</a></i>;<br/><a NAME="E2:88_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> implies ( <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> ) ) &amp; ( <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> implies <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> ) &amp; ( <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> implies ( <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> ) ) &amp; ( <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> implies <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> ) &amp; ( <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> implies ( <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> ) ) &amp; ( <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> implies <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> ) )
 <b>by </b><i><a class="ref" href="chain_1.html#T52" target="_self" title="CHAIN_1:th.52">Th52</a></i>;<br/><b>hence </b><a NAME="E3:88_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> )
 <b>by </b><i><a class="txt" href="chain_1.html#E1:88_1"><i><font color="Green" title="E61">A1</font></i></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></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:88"/>
 <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>
 <b>by </b><i><a class="ref" href="subset_1.html#T8" title="SUBSET_1:th.8">SUBSET_1:8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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