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<b>consider </b><font color="Maroon">c<sub>3</sub></font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> </span>)</span><b> such that </b><br/><a NAME="E2:21_1_1"/><i><font color="Green">E12</font></i>: 
 <a href="polynom3.html#K6" target="_self">Decomp</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="triang_1.html#K2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom3.html#K5" target="_self">TuplesOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E3:21_1_1"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> iff  <a href="wsierp_1.html#K8">Sum</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="polynom3.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E4:21_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span> <a href="polynom3.html#K3" target="_self">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0 as    <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
 <a href="wsierp_1.html#K8">Sum</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T104">RVSUM_1:104</a></i>
<br/>.= 
0 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T111">RVSUM_1:111</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <b>by </b><i><a class="txt" href="polynom3.html#E3:21_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<a NAME="E9:21_1_1"/>
( not  <a href="triang_1.html#K2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom3.html#K5" target="_self">TuplesOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="triang_1.html#K2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom3.html#K5" target="_self">TuplesOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font> is <a href="finset_1.html#V1">finite</a> )
 
;<br/>
<b>hence </b><a NAME="E10:21_1_1"/>
( not  <a href="polynom3.html#K6" target="_self">Decomp</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="polynom3.html#K6" target="_self">Decomp</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="polynom3.html#K6" target="_self">Decomp</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> is <a href="matrlin.html#V1">FinSequence-yielding</a> )
 <b>by </b><i><a class="txt" href="polynom3.html#E2:21_1_1"><i><font color="Green">E12</font></i></a></i>;<br/>


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