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<span class="kw">let </span><font color="Maroon" title="c1">R</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#V7" title="ALGSTR_0:attr.7">add-cancelable</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>   <a href="algstr_1.html#V1" title="ALGSTR_1:attr.1">left_zeroed</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</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">I</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a>   <a href="ideal_1.html#V3" title="IDEAL_1:attr.3">right-ideal</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b2">a</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">I</font><br/> for <font color="Olive" title="b3">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Olive" title="b3">n</font> <a href="binom.html#K5" title="BINOM:func.5">*</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">I</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">I</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a>   <a href="ideal_1.html#V3" title="IDEAL_1:attr.3">right-ideal</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</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">a</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">I</font><br/> for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Olive" title="b2">n</font> <a href="binom.html#K5" title="BINOM:func.5">*</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">a</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">I</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">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Olive" title="b1">n</font> <a href="binom.html#K5" title="BINOM:func.5">*</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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"> <font color="Maroon" title="c4">n</font> <a href="binom.html#K5" title="BINOM:func.5">*</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font></span><br/>







<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span>$1 <a href="binom.html#K5" title="BINOM:func.5">*</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>;<br/>
<a NAME="E1:30"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="lab"><font color="Green" title="E17">A2</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binom.html#K5" title="BINOM:func.5">*</a> <font color="Maroon" title="c3">a</font> = 
<span class="p1">(<span class="default">1 <a href="binom.html#K5" title="BINOM:func.5">*</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="binom.html#K5" title="BINOM:func.5">*</a> <font color="Maroon" title="c3">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binom.html#T16" title="BINOM:th.16">BINOM:16</a></span>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="binom.html#K5" title="BINOM:func.5">*</a> <font color="Maroon" title="c3">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binom.html#T14" title="BINOM:th.14">BINOM:14</a></span>
;<br/>
<span class="kw">assume </span><a NAME="E2:30_1"/>
<font color="Maroon" title="c5">n</font> <a href="binom.html#K5" title="BINOM:func.5">*</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</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">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<span class="kw">hence </span><a NAME="E3:30_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="ideal_1.html#D1" target="_self" title="IDEAL_1:def.1">Def1</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>
<a NAME="E2:30"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binom.html#K5" title="BINOM:func.5">*</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binom.html#T13" title="BINOM:th.13">BINOM:13</a></span>;<br/>
<a NAME="E3:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A3</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ideal_1.html#T3" target="_self" title="IDEAL_1:th.3">Th3</a></span>;<br/>
<a NAME="E4:30"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E3:30"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E16">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E5:30"/>
<font color="Maroon" title="c4">n</font> <a href="binom.html#K5" title="BINOM:func.5">*</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 ; <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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