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<a NAME="E1:153_1_1"/><i><font color="Green">E41</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20" target="_self">divisors</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:153_1_1"/>
not  <a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="153_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:153_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span>
 ;<br/><b>assume </b><a NAME="E2:153_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">bm1</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">bm2</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E4:153_1_1_1"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">bm1</font>,<font color="Maroon">bm2</font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E5:153_1_1_1"/>
<font color="Maroon">b</font> <a href="pboole.html#R6">=</a> <font color="Maroon">bm1</font> <a href="polynom1.html#K9" target="_self">+</a> <font color="Maroon">bm2</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T14" target="_self">Th14</a></i>;<br/><b>assume </b><a NAME="E6:153_1_1_1"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">m</font>
 ;<br/><b>then </b><span class="p1">(<span class="default"><a href="polynom1.html#K20" target="_self">divisors</a> <font color="Maroon">b</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> = 
<font color="Maroon">bm1</font>
<b>by </b><i><a class="ref" href="polynom1.html#T18" target="_self">Th18</a>, <a class="ref" href="polynom1.html#D3" target="_self">Def3</a>, <a class="ref" href="polynom1.html#T16" target="_self">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K20" target="_self">divisors</a> <font color="Maroon">b</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="polynom1.html#D2" target="_self">Def2</a>, <a class="ref" href="polynom1.html#D3" target="_self">Def3</a>, <a class="ref" href="polynom1.html#T16" target="_self">Th16</a></i>
;<br/><b>hence </b><a NAME="E8:153_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T17" target="_self">Th17</a>, <a class="ref" href="polynom1.html#T18" target="_self">Th18</a>, <a class="ref" href="polynom1.html#D2" target="_self">Def2</a>, <a class="ref" href="partfun2.html#T17">PARTFUN2:17</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:153_1_1"/>
 <a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="partfun2.html#T16">PARTFUN2:16</a></i>;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="matrlin.html#D1">MATRLIN:def 1</a><br/>

<b>assume </b><a NAME="E5:153_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K20" target="_self">divisors</a> <font color="Maroon">b</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> <b>by </b><i><a class="ref" href="polynom1.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E8:153_1_1"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">p</font>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K10" target="_self">-'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="polynom1.html#T18" target="_self">Th18</a>, </i>;<br/>
<b>hence </b><a NAME="E9:153_1_1"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> is   <a href="finseq_1.html#NM1">FinSequence</a>
 <b>by </b><i><a class="ref" href="polynom1.html#T18" target="_self">Th18</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>


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