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<b>let </b><font color="Maroon">p</font> be   <a href="polynom1.html#NM3">Series</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>, <font color="Maroon">q</font> be   <a href="polynom1.html#NM3">Series</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>;<br/>

<b>reconsider </b><font color="Maroon">pq</font> = <font color="Maroon">p</font> <a href="polynom1.html#K28" target="_self">*'</a> <font color="Maroon">q</font>, <font color="Maroon">qp</font> = <font color="Maroon">q</font> <a href="polynom1.html#K28" target="_self">*'</a> <font color="Maroon">p</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="175_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be    <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font>;<br/><b>consider </b><font color="Maroon">s</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E2:175_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">pq</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E3:175_1_1_1"/><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span>
 <b>and </b><br/><a NAME="E4:175_1_1_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s</font> holds <br/> ex <font color="Olive">b1</font>, <font color="Olive">b2</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> st <br/>( <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="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">b1</font>,<font color="Olive">b2</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Olive">b1</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Olive">b2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="polynom1.html#T24" target="_self">Th24</a></i>;<br/><b>consider </b><font color="Maroon">t</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:175_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">qp</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">t</font>
 <b>and </b><br/><a NAME="E7:175_1_1_1"/><i><font color="Green">E45</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span>
 <b>and </b><br/><a NAME="E8:175_1_1_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">t</font> holds <br/> ex <font color="Olive">b1</font>, <font color="Olive">b2</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> st <br/>( <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="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">b1</font>,<font color="Olive">b2</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">t</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Olive">b1</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Olive">b2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="polynom1.html#T24" target="_self">Th24</a></i>;<br/><a NAME="E9:175_1_1_1"/><i><font color="Green">E47</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s</font> <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> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">t</font> <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><a class="ref" href="polynom1.html#T29" target="_self">Th29</a>, , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><b>then reconsider </b><font color="Maroon">ds</font> =  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">b1</font>, <font color="Olive">b2</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> st <br/>( <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">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">b1</font>,<font color="Olive">b2</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <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">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">b2</font>,<font color="Olive">b1</font></span><a href="finseq_1.html#K10">*&gt;</a></span> );<br/><div><i><font color="Green">E69</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="175_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:175_1_1_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ds</font>
 ;<br/><b>then consider </b><font color="Maroon">b1</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">b2</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:175_1_1_1_1"/><i><font color="Green">E71</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">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b1</font>,<font color="Maroon">b2</font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E4:175_1_1_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">b</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1</font> <a href="polynom1.html#K9" target="_self">+</a> <font color="Maroon">b2</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#T14" target="_self">Th14</a></i>;<br/><b>consider </b><font color="Maroon">d</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:175_1_1_1_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">d</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>
 <b>and </b><br/><a NAME="E7:175_1_1_1_1"/><i><font color="Green">E109</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">d</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b2</font>,<font color="Maroon">b1</font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="polynom1.html#E14:175_1_1_1"><i><font color="Green">E111</font></i></a>, <a class="ref" href="polynom1.html#T15" target="_self">Th15</a></i>;<br/><b>reconsider </b><font color="Maroon">d</font> = <font color="Maroon">d</font> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">d</font> = <font color="Maroon">d</font>;<br/><b>thus </b><a NAME="E9:175_1_1_1_1"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ds</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T29" target="_self">Th29</a>, <a class="txt" href="polynom1.html#E15:175_1_1_1"><i><font color="Green">E112</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><b>thus </b><a NAME="E10:175_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">e</font>,<font color="Maroon">d</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="175_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">b1</font>
;<br/>

<b>take </b>
<font color="Maroon">b2</font>
;<br/>

<b>thus </b><a NAME="E1:175_1_1_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="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b1</font>,<font color="Maroon">b2</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <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">d</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b2</font>,<font color="Maroon">b1</font></span><a href="finseq_1.html#K10">*&gt;</a></span> )
 <b>by </b><i>, <a class="txt" href="polynom1.html#E7:175_1_1_1_1"><i><font color="Green">E109</font></i></a>, <a class="txt" href="polynom1.html#E13:175_1_1_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="polynom1.html#E15:175_1_1_1"><i><font color="Green">E112</font></i></a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">ds</font>,<font color="Maroon">ds</font><b> such that </b><br/><a NAME="E13:175_1_1_1"/><i><font color="Green">E110</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ds</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">e</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>();<br/></i><a NAME="E14:175_1_1_1"/><i><font color="Green">E111</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ds</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">ds</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><a NAME="E15:175_1_1_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="175_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:175_1_1_1_2"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E2:175_1_1_1_2"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E3:175_1_1_1_2"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 ;<br/>

<b>consider </b><font color="Maroon">b1</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">b2</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E5:175_1_1_1_2"/><i><font color="Green">E116</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="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b1</font>,<font color="Maroon">b2</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <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> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b2</font>,<font color="Maroon">b1</font></span><a href="finseq_1.html#K10">*&gt;</a></span> )
 <b>by </b><i>, <a class="txt" href="polynom1.html#E15:175_1_1_1"><i><font color="Green">E112</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">b3</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">b4</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E7:175_1_1_1_2"/><i><font color="Green">E117</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="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b3</font>,<font color="Maroon">b4</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <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> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b4</font>,<font color="Maroon">b3</font></span><a href="finseq_1.html#K10">*&gt;</a></span> )
 <b>by </b><i>, <a class="txt" href="polynom1.html#E1:175_1_1_1_2"><i><font color="Green">E113</font></i></a></i>;<br/>
<a NAME="E8:175_1_1_1_2"/>
( <font color="Maroon">b2</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b4</font> &amp; <font color="Maroon">b1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b3</font> )
 
<b>by </b><i><a class="txt" href="polynom1.html#E2:175_1_1_1_2"><i><font color="Green">E114</font></i></a>, <a class="txt" href="polynom1.html#E3:175_1_1_1_2"><i><font color="Green">E115</font></i></a>, <a class="txt" href="polynom1.html#E5:175_1_1_1_2"><i><font color="Green">E116</font></i></a>, <a class="ref" href="group_7.html#T2">GROUP_7:2</a></i>;<br/>
<b>hence </b><a NAME="E9:175_1_1_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, <a class="txt" href="polynom1.html#E15:175_1_1_1"><i><font color="Green">E112</font></i></a>, <a class="txt" href="polynom1.html#E1:175_1_1_1_2"><i><font color="Green">E113</font></i></a>, <a class="txt" href="polynom1.html#E3:175_1_1_1_2"><i><font color="Green">E115</font></i></a>, <a class="txt" href="polynom1.html#E5:175_1_1_1_2"><i><font color="Green">E116</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E16:175_1_1_1"/>
<font color="Maroon">ds</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="175_1_1_1_3"><b>proof </b></a><div class="add">

<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="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:175_1_1_1_3"/><i><font color="Green">E194</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ds</font>
 ;<br/>

<a NAME="E2:175_1_1_1_3"/><b>then </b><i><font color="Green">E195</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="polynom1.html#E13:175_1_1_1"><i><font color="Green">E110</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E3:175_1_1_1_3"/><b>then </b><i><font color="Green">E196</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="polynom1.html#E13:175_1_1_1"><i><font color="Green">E110</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">b1</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">b2</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E5:175_1_1_1_3"/><i><font color="Green">E199</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="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b1</font>,<font color="Maroon">b2</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <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> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b2</font>,<font color="Maroon">b1</font></span><a href="finseq_1.html#K10">*&gt;</a></span> )
 <b>by </b><i>, <a class="txt" href="polynom1.html#E15:175_1_1_1"><i><font color="Green">E112</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">b3</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">b4</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E7:175_1_1_1_3"/><i><font color="Green">E201</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="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b3</font>,<font color="Maroon">b4</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <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> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b4</font>,<font color="Maroon">b3</font></span><a href="finseq_1.html#K10">*&gt;</a></span> )
 <b>by </b><i>, <a class="txt" href="polynom1.html#E13:175_1_1_1"><i><font color="Green">E110</font></i></a>, </i>;<br/>
<a NAME="E8:175_1_1_1_3"/>
( <font color="Maroon">b3</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">b4</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1</font> )
 
<b>by </b><i>, , <a class="ref" href="group_7.html#T2">GROUP_7:2</a></i>;<br/>
<b>hence </b><a NAME="E9:175_1_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="txt" href="polynom1.html#E13:175_1_1_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="polynom1.html#E15:175_1_1_1"><i><font color="Green">E112</font></i></a>, , , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E17:175_1_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ds</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E13:175_1_1_1"><i><font color="Green">E110</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s</font> <b>by </b><i><a class="txt" href="polynom1.html#E14:175_1_1_1"><i><font color="Green">E111</font></i></a>, <a class="ref" href="funct_2.html#T83">FUNCT_2:83</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="175_1_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:175_1_1_1_4"/><i><font color="Green">E202</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">t</font>
 ;<br/><a NAME="E2:175_1_1_1_4"/><b>then </b><i><font color="Green">E204</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="txt" href="polynom1.html#E13:175_1_1_1"><i><font color="Green">E110</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E3:175_1_1_1_4"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ds</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E13:175_1_1_1"><i><font color="Green">E110</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">fi</font> = <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><b>consider </b><font color="Maroon">b1</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">b2</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E6:175_1_1_1_4"/><i><font color="Green">E205</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">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b1</font>,<font color="Maroon">b2</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">t</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b1</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="polynom1.html#T32" target="_self">Th32</a>, <a class="txt" href="polynom1.html#E15:175_1_1_1"><i><font color="Green">E112</font></i></a></i>;<br/><b>consider </b><font color="Maroon">b3</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">b4</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E8:175_1_1_1_4"/><i><font color="Green">E206</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">fi</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b3</font>,<font color="Maroon">b4</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">fi</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b3</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b4</font></span>)</span> )
 <b>by </b><i><a class="ref" href="polynom1.html#T30" target="_self">Th30</a>, <a class="txt" href="polynom1.html#E13:175_1_1_1"><i><font color="Green">E110</font></i></a>, </i>;<br/><b>consider </b><font color="Maroon">b5</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">b6</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E10:175_1_1_1_4"/><i><font color="Green">E207</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="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b5</font>,<font color="Maroon">b6</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <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> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b6</font>,<font color="Maroon">b5</font></span><a href="finseq_1.html#K10">*&gt;</a></span> )
 <b>by </b><i>, , <a class="txt" href="polynom1.html#E15:175_1_1_1"><i><font color="Green">E112</font></i></a></i>;<br/><a NAME="E11:175_1_1_1_4"/><i><font color="Green">E208</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T29" target="_self">Th29</a>, , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E12:175_1_1_1_4"/><i><font color="Green">E211</font></i>: 
( <font color="Maroon">t</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">fi</font> )
 <b>by </b><i><a class="txt" href="polynom1.html#E13:175_1_1_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="polynom1.html#E15:175_1_1_1"><i><font color="Green">E112</font></i></a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E13:175_1_1_1_4"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s</font> <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><a class="ref" href="polynom1.html#T29" target="_self">Th29</a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E14:175_1_1_1_4"/><b>then </b>
( <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">i</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="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <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">fi</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="funct_1.html#K1">.</a> <font color="Maroon">fi</font> )
 <b>by </b><i><a class="txt" href="polynom1.html#E13:175_1_1_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="polynom1.html#E15:175_1_1_1"><i><font color="Green">E112</font></i></a>, , , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E15:175_1_1_1_4"/><b>then </b>
( <font color="Maroon">b1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b5</font> &amp; <font color="Maroon">b2</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b6</font> &amp; <font color="Maroon">b3</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b6</font> &amp; <font color="Maroon">b4</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b5</font> )
 <b>by </b><i>, , , <a class="ref" href="group_7.html#T2">GROUP_7:2</a></i>;<br/><b>hence </b><a NAME="E16:175_1_1_1_4"/>
<font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="polynom1.html#T33" target="_self">Th33</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E20:175_1_1_1"/>
<font color="Maroon">pq</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">qp</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#T29" target="_self">Th29</a>, <a class="ref" href="polynom1.html#T31" target="_self">Th31</a>, , <a class="ref" href="rlvect_2.html#T8">RLVECT_2:8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:175_1_1"/>
<font color="Maroon">p</font> <a href="polynom1.html#K28" target="_self">*'</a> <font color="Maroon">q</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">q</font> <a href="polynom1.html#K28" target="_self">*'</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


</div>
