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<b>consider </b><font color="Maroon" title="c3">A</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span><b> such that </b><br/><a NAME="E2:21_1_1"/><i><font color="Green" title="E12">A1</font></i>: 
 <a href="polynom3.html#K6" title="POLYNOM3:func.6">Decomp</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom3.html#K5" title="POLYNOM3:func.5">TuplesOrder</a> <font color="Maroon" title="c1">i</font></span>)</span>,<font color="Maroon" title="c3">A</font>
 <b>and </b><br/><a NAME="E3:21_1_1"/><i><font color="Green" title="E13">A2</font></i>: 
for <font color="Olive" title="b1">p</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">i</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> iff  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> )
 <b>by </b><i><a class="ref" href="polynom3.html#D4" target="_self" title="POLYNOM3:def.4">Def4</a></i>;<br/>
<a NAME="E4:21_1_1"/>
<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">p1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="polynom3.html#K3" title="POLYNOM3:func.3">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">n</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">i</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">p2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
 <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c4">p1</font> = 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c5">p2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T104" title="RVSUM_1:th.104">RVSUM_1:104</a></i>
<br/>.= 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T111" title="RVSUM_1:th.111">RVSUM_1:111</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">A</font> = <font color="Maroon" title="c3">A</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <b>by </b><i><a class="txt" href="polynom3.html#E3:21_1_1"><i><font color="Green" title="E13">A2</font></i></a></i>;<br/>
<a NAME="E9:21_1_1"/>
( not  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom3.html#K5" title="POLYNOM3:func.5">TuplesOrder</a> <font color="Maroon" title="c1">i</font></span>)</span>,<font color="Maroon" title="c6">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom3.html#K5" title="POLYNOM3:func.5">TuplesOrder</a> <font color="Maroon" title="c1">i</font></span>)</span>,<font color="Maroon" title="c6">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 
;<br/>
<b>hence </b><a NAME="E10:21_1_1"/>
( not  <a href="polynom3.html#K6" title="POLYNOM3:func.6">Decomp</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">i</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="polynom3.html#K6" title="POLYNOM3:func.6">Decomp</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">i</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="polynom3.html#K6" title="POLYNOM3:func.6">Decomp</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">i</font> is <a href="matrlin.html#V1" title="MATRLIN:attr.1">FinSequence-yielding</a> )
 <b>by </b><i><a class="txt" href="polynom3.html#E2:21_1_1"><i><font color="Green" title="E12">A1</font></i></a></i>;<br/>


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