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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">C</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 <br/> ( <font color="Olive" title="b3">C</font> is    <a href="chain_1.html#M3" title="CHAIN_1:mode.3">Cycle</a> of <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1,<font color="Olive" title="b2">G</font> iff  <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <font color="Olive" title="b3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="chain_1.html#K5" title="CHAIN_1:func.5">0_</a> <font color="Maroon" title="c1">k</font>,<font color="Olive" title="b2">G</font> )</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">C</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 <br/> ( <font color="Olive" title="b2">C</font> is    <a href="chain_1.html#M3" title="CHAIN_1:mode.3">Cycle</a> of <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1,<font color="Olive" title="b1">G</font> iff  <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <font color="Olive" title="b2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="chain_1.html#K5" title="CHAIN_1:func.5">0_</a> <font color="Maroon" title="c1">k</font>,<font color="Olive" title="b1">G</font> )</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">C</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 <br/> ( <font color="Olive" title="b1">C</font> is    <a href="chain_1.html#M3" title="CHAIN_1:mode.3">Cycle</a> of <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1,<font color="Maroon" title="c3">G</font> iff  <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="chain_1.html#K5" title="CHAIN_1:func.5">0_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">G</font> )</span><br/><b>let </b><font color="Maroon" title="c4">C</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"> ( <font color="Maroon" title="c4">C</font> is    <a href="chain_1.html#M3" title="CHAIN_1:mode.3">Cycle</a> of <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1,<font color="Maroon" title="c3">G</font> iff  <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="chain_1.html#K5" title="CHAIN_1:func.5">0_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">G</font> )</span><br/>







<div><b>hereby </b> <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> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="chain_1.html#K5" title="CHAIN_1:func.5">0_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">G</font> implies <font color="Maroon" title="c4">C</font> is    <a href="chain_1.html#M3" title="CHAIN_1:mode.3">Cycle</a> of <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1,<font color="Maroon" title="c3">G</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:92_1"/>
<font color="Maroon" title="c4">C</font> is    <a href="chain_1.html#M3" title="CHAIN_1:mode.3">Cycle</a> of <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1,<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> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="chain_1.html#K5" title="CHAIN_1:func.5">0_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">G</font></span><br/><b>then consider </b><font color="Maroon" title="c5">k'</font> being    <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> <b> such that </b><br/><a NAME="E3:92_1"/><i><font color="Green" title="E65">A1</font></i>: 
( <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  ex <font color="Olive" title="b1">C'</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="c5">k'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c3">G</font> st <br/>( <font color="Olive" title="b1">C'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> &amp;  <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <font color="Olive" title="b1">C'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="chain_1.html#K5" title="CHAIN_1:func.5">0_</a> <font color="Maroon" title="c5">k'</font>,<font color="Maroon" title="c3">G</font> ) )
 <b>by </b><i><a class="ref" href="chain_1.html#D15" target="_self" title="CHAIN_1:def.15">Def15</a></i>;<br/><b>thus </b><a NAME="E4:92_1"/>
 <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="chain_1.html#K5" title="CHAIN_1:func.5">0_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">G</font>
 <b>by </b><i><a class="txt" href="chain_1.html#E3:92_1"><i><font color="Green" title="E65">A1</font></i></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>thus </b><a NAME="E2:92"/>
(  <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="chain_1.html#K5" title="CHAIN_1:func.5">0_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">G</font> implies <font color="Maroon" title="c4">C</font> is    <a href="chain_1.html#M3" title="CHAIN_1:mode.3">Cycle</a> of <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1,<font color="Maroon" title="c3">G</font> )
 <b>by </b><i><a class="ref" href="chain_1.html#D15" target="_self" title="CHAIN_1:def.15">Def15</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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