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<a NAME="E1:137_1_1"/><i><font color="Green" title="E81">A1</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#D19" target="_self" title="POLYNOM1:def.19">Def19</a></i>;<br/>
<b>hence </b><a NAME="E2:137_1_1"/>
not  <a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font> is <a href="matrlin.html#V1" title="MATRLIN:attr.1">FinSequence-yielding</a> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="137_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">k</font>, <font color="Maroon" title="c4">m</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> &amp; <font color="Maroon" title="c4">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c4">m</font> implies <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> )</span><br/><b>assume </b><a NAME="E1:137_1_1_1"/><i><font color="Green" title="E82">A2</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c4">m</font> implies <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> )</span><br/><b>assume </b><a NAME="E2:137_1_1_1"/><i><font color="Green" title="E83">A3</font></i>: 
<font color="Maroon" title="c4">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c4">m</font> implies <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> )</span><br/><b>then consider </b><font color="Maroon" title="c5">bm1</font>, <font color="Maroon" title="c6">bm2</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E4:137_1_1_1"/><i><font color="Green" title="E84">A4</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">bm1</font>,<font color="Maroon" title="c6">bm2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E5:137_1_1_1"/>
<font color="Maroon" title="c2">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c5">bm1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c6">bm2</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T72" target="_self" title="POLYNOM1:th.72">Th72</a></i>;<br/><b>assume </b><a NAME="E6:137_1_1_1"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c4">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font></span><br/><b>then </b><span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> = 
<font color="Maroon" title="c5">bm1</font>
<b>by </b><i><a class="txt" href="#E1:137_1_1_1"><i><font color="Green" title="E82">A2</font></i></a>, <a class="txt" href="#E4:137_1_1_1"><i><font color="Green" title="E84">A4</font></i></a>, <a class="ref" href="polynom1.html#T74" target="_self" title="POLYNOM1:th.74">Th74</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">m</font>
<b>by </b><i><a class="txt" href="#E2:137_1_1_1"><i><font color="Green" title="E83">A3</font></i></a>, <a class="txt" href="#E4:137_1_1_1"><i><font color="Green" title="E84">A4</font></i></a>, <a class="ref" href="polynom1.html#T74" target="_self" title="POLYNOM1:th.74">Th74</a></i>
;<br/><b>hence </b><a NAME="E8:137_1_1_1"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font>
 <b>by </b><i><a class="txt" href="#E1:137_1_1"><i><font color="Green" title="E81">A1</font></i></a>, <a class="txt" href="#E1:137_1_1_1"><i><font color="Green" title="E82">A2</font></i></a>, <a class="txt" href="#E2:137_1_1_1"><i><font color="Green" title="E83">A3</font></i></a>, <a class="ref" href="partfun2.html#T17" title="PARTFUN2:th.17">PARTFUN2:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:137_1_1"/>
 <a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="partfun2.html#T16" title="PARTFUN2:th.16">PARTFUN2:16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font> is <a href="matrlin.html#V1" title="MATRLIN:attr.1">FinSequence-yielding</a></span><br/>

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="matrlin.html#D1" title="MATRLIN:def.1">MATRLIN:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> or <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> is    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  )</span><br/>

<b>assume </b><a NAME="E5:137_1_1"/><i><font color="Green" title="E82">A5</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> is    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> </span><br/>

<b>then reconsider </b><font color="Maroon" title="c4">k</font> = <font color="Maroon" title="c3">x</font> as    <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>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c5">p</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">k</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" target="_self" title="POLYNOM1:def.14">Def14</a></i>;<br/>
<a NAME="E8:137_1_1"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-'</a> <font color="Maroon" title="c5">p</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="#E5:137_1_1"><i><font color="Green" title="E82">A5</font></i></a>, <a class="ref" href="polynom1.html#D19" target="_self" title="POLYNOM1:def.19">Def19</a></i>;<br/>
<b>hence </b><a NAME="E9:137_1_1"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> is   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>
 <b>by </b><i><a class="txt" href="#E5:137_1_1"><i><font color="Green" title="E82">A5</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>; <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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