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<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <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">L</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="vectsp_1.html#V1" title="VECTSP_1:attr.1">right-distributive</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> <br/> for <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font> being  <a href="polynom1.html#NM3" title="POLYNOM1:NM.3">Series</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font><br/> for <font color="Olive" title="b4">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> holds  <font color="Olive" title="b4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Olive" title="b3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Olive" title="b2">p</font></span>)</span> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Olive" title="b3">q</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="vectsp_1.html#V1" title="VECTSP_1:attr.1">right-distributive</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <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">p</font>, <font color="Olive" title="b2">q</font> being  <a href="polynom1.html#NM3" title="POLYNOM1:NM.3">Series</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font><br/> for <font color="Olive" title="b3">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> holds  <font color="Olive" title="b3">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Olive" title="b2">q</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c4">p2</font> be   <a href="polynom1.html#NM3" title="POLYNOM1:NM.3">Series</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>; <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">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> holds  <font color="Olive" title="b1">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p1</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Maroon" title="c4">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p2</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c5">b</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">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="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p1</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Maroon" title="c4">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p2</font></span>)</span></span><br/>







<b>set </b><font color="Maroon" title="c6">q1</font> = <font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p1</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Maroon" title="c4">p2</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">q2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p2</font></span>)</span>;<br/>
<i><font color="Green" title="E9">A1</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p1</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Maroon" title="c4">p2</font></span>)</span></span>)</span> = 
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">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="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p1</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Maroon" title="c4">p2</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p1</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Maroon" title="c4">p2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> )</span><br/><b>assume </b><a NAME="E1:20_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p1</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Maroon" title="c4">p2</font></span>)</span></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"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p1</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Maroon" title="c4">p2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span><br/><b>then reconsider </b><font color="Maroon" title="c9">u</font> = <font color="Maroon" title="c8">x</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" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_2_1_1"/>
( <font color="Maroon" title="c5">b</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c9">u</font> or  not <font color="Maroon" title="c5">b</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c9">u</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_2_1_1_1"><b>case </b></a><a NAME="E1:20_2_1_1_1"/><i><font color="Green" title="E10">A2</font></i>: 
<font color="Maroon" title="c5">b</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c9">u</font>
 ; <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"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p1</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Maroon" title="c4">p2</font></span>)</span></span>)</span> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c9">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p2</font></span>)</span></span>)</span> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c9">u</font></span><br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p1</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Maroon" title="c4">p2</font></span>)</span></span>)</span> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c9">u</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p1</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Maroon" title="c4">p2</font></span>)</span> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">u</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-'</a> <font color="Maroon" title="c5">b</font></span>)</span>
<b>by </b><i><a class="ref" href="polyred.html#D1" title="POLYRED:def.1">POLYRED:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p1</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">u</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-'</a> <font color="Maroon" title="c5">b</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p2</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">u</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-'</a> <font color="Maroon" title="c5">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D21" title="POLYNOM1:def.21">POLYNOM1:def 21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c9">u</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p2</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">u</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-'</a> <font color="Maroon" title="c5">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:20_2_1_1_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="polyred.html#D1" title="POLYRED:def.1">POLYRED:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c9">u</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p2</font></span>)</span> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c9">u</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:20_2_1_1_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="polyred.html#D1" title="POLYRED:def.1">POLYRED:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p2</font></span>)</span></span>)</span> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c9">u</font>
<b>by </b><i><a class="ref" href="polynom1.html#D21" title="POLYNOM1:def.21">POLYNOM1:def 21</a></i>
;<br/> <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_2_1_1_2"><b>case </b></a><a NAME="E1:20_2_1_1_2"/><i><font color="Green" title="E10">A3</font></i>: 
not <font color="Maroon" title="c5">b</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c9">u</font>
 ; <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"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p1</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Maroon" title="c4">p2</font></span>)</span></span>)</span> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c9">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p2</font></span>)</span></span>)</span> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c9">u</font></span><br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p1</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Maroon" title="c4">p2</font></span>)</span></span>)</span> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c9">u</font> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
<b>by </b><i><a class="ref" href="polyred.html#D1" title="POLYRED:def.1">POLYRED:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c9">u</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:20_2_1_1_2"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="polyred.html#D1" title="POLYRED:def.1">POLYRED:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c9">u</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p2</font></span>)</span> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c9">u</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:20_2_1_1_2"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="polyred.html#D1" title="POLYRED:def.1">POLYRED:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p2</font></span>)</span></span>)</span> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c9">u</font>
<b>by </b><i><a class="ref" href="polynom1.html#D21" title="POLYNOM1:def.21">POLYNOM1:def 21</a></i>
;<br/> <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></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:20_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p1</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Maroon" title="c4">p2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
 ; <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="E3:20"/>
<font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p1</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Maroon" title="c4">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p2</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:20"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</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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