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<b>let </b><font color="Maroon" title="c1">n</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">G</font> being  <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a><br/> for <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">G</font> holds   <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <font color="Olive" title="b2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c1">n</font></span><br/><b>let </b><font color="Maroon" title="c2">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</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">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> holds   <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c1">n</font></span><br/><b>let </b><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c1">n</font></span><br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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>means </b> <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default">$1 <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> $1;<br/>
<a NAME="E1:16"/><i><font color="Green" title="E10">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="txt" href="group_4.html#E18"><i><font color="Green" title="E8">Lm1</font></i></a></i>;<br/>
<a NAME="E2:16"/><i><font color="Green" title="E11">A2</font></i>: 
for <font color="Olive" title="b1">n</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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<b>by </b><i><a class="txt" href="group_4.html#E19"><i><font color="Green" title="E9">Lm2</font></i></a></i>;<br/>
<a NAME="E3:16"/>
for <font color="Olive" title="b1">n</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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="group_4.html#E1:16"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="group_4.html#E2:16"><i><font color="Green" title="E11">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:16"/>
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c1">n</font>
 ; <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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