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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>  st <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">d</font> holds <br/>for <font color="Olive" title="b3">C</font> being  <a href="chain_1.html#NM4" title="CHAIN_1:NM.4">Chain</a> of <font color="Maroon" title="c1">k</font>,<font color="Olive" title="b2">G</font> holds  <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>,<font color="Olive" title="b2">G</font></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>  st <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">d</font> holds <br/>for <font color="Olive" title="b2">C</font> being  <a href="chain_1.html#NM4" title="CHAIN_1:NM.4">Chain</a> of <font color="Maroon" title="c1">k</font>,<font color="Olive" title="b1">G</font> holds  <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>,<font color="Olive" title="b1">G</font></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"> ( <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">d</font> implies for <font color="Olive" title="b1">C</font> being  <a href="chain_1.html#NM4" title="CHAIN_1:NM.4">Chain</a> of <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">G</font> holds  <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>,<font color="Maroon" title="c3">G</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:91"/><span class="lab"><font color="Green" title="E63">A1</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <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">C</font> being  <a href="chain_1.html#NM4" title="CHAIN_1:NM.4">Chain</a> of <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">G</font> holds  <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>,<font color="Maroon" title="c3">G</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">C</font> be   <a href="chain_1.html#NM4" title="CHAIN_1:NM.4">Chain</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="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>,<font color="Maroon" title="c3">G</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">k9</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:91"/><span class="lab"><font color="Green" title="E64">A2</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k9</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">k9</font> = <font color="Maroon" title="c5">k9</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">C9</font> = <font color="Maroon" title="c4">C</font> as   <a href="chain_1.html#NM4" title="CHAIN_1:NM.4">Chain</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">k9</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c3">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:91"><span class="lab"><font color="Green" title="E64">A2</font></span></a></span>;<br/>
<a NAME="E6:91"/>
 <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <font color="Maroon" title="c7">C9</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="c6">k9</font>,<font color="Maroon" title="c3">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E3:91"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="ref" href="chain_1.html#T53" target="_self" title="CHAIN_1:th.53">Th53</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:91"/>
<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>,<font color="Maroon" title="c3">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:91"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="ref" href="chain_1.html#D15" target="_self" title="CHAIN_1:def.15">Def15</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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