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<b>set </b><font color="Maroon" title="c3">PRPNR</font> =  <a href="polynom3.html#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <span class="p1">(<span class="default"><a href="polynom1.html#K28" title="POLYNOM1:func.28">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c4">PN1R</font> =  <a href="polynom1.html#K28" title="POLYNOM1:func.28">Polynom-Ring</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">R</font>;<br/>
<b>set </b><font color="Maroon" title="c5">PNR</font> =  <a href="polynom1.html#K28" title="POLYNOM1:func.28">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font>;<br/>
<b>set </b><font color="Maroon" title="c6">CPN1R</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K28" title="POLYNOM1:func.28">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span>;<br/>
<b>let </b><font color="Maroon" title="c7">A</font>, <font color="Maroon" title="c8">B</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K28" title="POLYNOM1:func.28">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="polynom3.html#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <span class="p2">(<span class="default"><a href="polynom1.html#K28" title="POLYNOM1:func.28">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">p1</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b2">p2</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b3">p3</font> being  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K28" title="POLYNOM1:func.28">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span><br/> for <font color="Olive" title="b4">i</font> being   <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/> for <font color="Olive" title="b5">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">n</font>  st <font color="Olive" title="b3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">i</font> holds <br/><font color="Olive" title="b2">p2</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Olive" title="b5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p1</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">b</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Olive" title="b4">i</font></span>)</span> ) &amp; ( for <font color="Olive" title="b1">p1</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b2">p2</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b3">p3</font> being  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K28" title="POLYNOM1:func.28">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span><br/> for <font color="Olive" title="b4">i</font> being   <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/> for <font color="Olive" title="b5">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">n</font>  st <font color="Olive" title="b3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">i</font> holds <br/><font color="Olive" title="b2">p2</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Olive" title="b5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p1</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">b</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Olive" title="b4">i</font></span>)</span> ) implies <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font> )</span><br/>

<b>assume </b><a NAME="E1:38_1_2"/><i><font color="Green" title="E33">A26</font></i>: 
for <font color="Olive" title="b1">p1</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b2">p2</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b3">p3</font> being  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K28" title="POLYNOM1:func.28">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span><br/> for <font color="Olive" title="b4">i</font> being   <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/> for <font color="Olive" title="b5">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">n</font>  st <font color="Olive" title="b3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">i</font> holds <br/><font color="Olive" title="b2">p2</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Olive" title="b5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p1</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">b</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Olive" title="b4">i</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">p1</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">R</font> ex <font color="Olive" title="b2">p2</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font> ex <font color="Olive" title="b3">p3</font> being  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K28" title="POLYNOM1:func.28">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> ex <font color="Olive" title="b4">i</font> being   <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>  ex <font color="Olive" title="b5">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">n</font> st <br/>( <font color="Olive" title="b3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">i</font> &amp; not <font color="Olive" title="b2">p2</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Olive" title="b5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p1</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">b</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Olive" title="b4">i</font></span>)</span> ) or <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font> )</span><br/>

<b>assume </b><a NAME="E2:38_1_2"/><i><font color="Green" title="E34">A27</font></i>: 
for <font color="Olive" title="b1">p1</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b2">p2</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b3">p3</font> being  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K28" title="POLYNOM1:func.28">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span><br/> for <font color="Olive" title="b4">i</font> being   <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/> for <font color="Olive" title="b5">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">n</font>  st <font color="Olive" title="b3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">i</font> holds <br/><font color="Olive" title="b2">p2</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Olive" title="b5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p1</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">b</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Olive" title="b4">i</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="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K28" title="POLYNOM1:func.28">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span> implies <font color="Maroon" title="c7">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> )</span><br/><b>assume </b><a NAME="E1:38_1_2_1"/><i><font color="Green" title="E35">A28</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K28" title="POLYNOM1:func.28">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">R</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="c7">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span><br/><b>then reconsider </b><font color="Maroon" title="c10">x''</font> = <font color="Maroon" title="c9">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K28" title="POLYNOM1:func.28">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span> ;<br/><b>reconsider </b><font color="Maroon" title="c11">x'</font> = <font color="Maroon" title="c9">x</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="#E1:38_1_2_1"><i><font color="Green" title="E35">A28</font></i></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">Ax</font> = <font color="Maroon" title="c7">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x''</font>, <font color="Maroon" title="c13">Bx</font> = <font color="Maroon" title="c8">B</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x''</font> as   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K28" title="POLYNOM1:func.28">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> <b>by </b><i><a class="ref" href="polynom3.html#D12" title="POLYNOM3:def.12">POLYNOM3:def 12</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  implies <font color="Maroon" title="c12">Ax</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Bx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> )</span><br/><b>assume </b><a NAME="E1:38_1_2_1_1"/>
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="c12">Ax</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Bx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span><br/><b>then reconsider </b><font color="Maroon" title="c15">i'</font> = <font color="Maroon" title="c14">i</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="c16">Axi</font> = <font color="Maroon" title="c12">Ax</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">i'</font>, <font color="Maroon" title="c17">Bxi</font> = <font color="Maroon" title="c13">Bx</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">i'</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c2">n</font> implies <font color="Maroon" title="c16">Axi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Bxi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b</font> )</span><br/><b>assume </b><a NAME="E1:38_1_2_1_1_1"/>
<font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c2">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c16">Axi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Bxi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b</font></span><br/><b>then reconsider </b><font color="Maroon" title="c19">b'</font> = <font color="Maroon" title="c18">b</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">n</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/><b>thus </b><font color="Maroon" title="c16">Axi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b</font> = 
<font color="Maroon" title="c11">x'</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">b'</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c15">i'</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:38_1_2"><i><font color="Green" title="E33">A26</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c17">Bxi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b</font>
<b>by </b><i><a class="txt" href="#E2:38_1_2"><i><font color="Green" title="E34">A27</font></i></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="E5:38_1_2_1_1"/>
<font color="Maroon" title="c12">Ax</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Bx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</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="E6:38_1_2_1"/>
<font color="Maroon" title="c7">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</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="E4:38_1_2"/>
<font color="Maroon" title="c7">A</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">B</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</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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