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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">T</font> being  <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a>  <a href="bagorder.html#NM1" title="BAGORDER:NM.1">TermOrder</a> of <font color="Maroon" title="c1">n</font><br/> for <font color="Olive" title="b2">L</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="struct_0.html#L2" title="STRUCT_0:struct.2">ZeroStr</a> <br/> for <font color="Olive" title="b3">p</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">Series</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">L</font><br/> for <font color="Olive" title="b4">b1</font>, <font color="Olive" title="b5">b2</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b4">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Olive" title="b5">b2</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Olive" title="b3">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b4">b1</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">b2</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Olive" title="b3">p</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">T</font> be   <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a>  <a href="bagorder.html#NM1" title="BAGORDER:NM.1">TermOrder</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="struct_0.html#L2" title="STRUCT_0:struct.2">ZeroStr</a> <br/> for <font color="Olive" title="b2">p</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">Series</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font><br/> for <font color="Olive" title="b3">b1</font>, <font color="Olive" title="b4">b2</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b3">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Olive" title="b4">b2</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Olive" title="b2">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b3">b1</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">b2</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Olive" title="b2">p</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">L</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="struct_0.html#L2" title="STRUCT_0:struct.2">ZeroStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">Series</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><br/> for <font color="Olive" title="b2">b1</font>, <font color="Olive" title="b3">b2</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b2">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Olive" title="b3">b2</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Olive" title="b1">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">b1</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">b2</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Olive" title="b1">p</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">p</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">Series</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">b1</font>, <font color="Olive" title="b2">b2</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Olive" title="b2">b2</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">b1</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b2</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">b1</font>, <font color="Maroon" title="c6">b2</font> be   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">b1</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b2</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font></span>)</span></span><br/>









<span class="kw">set </span><font color="Maroon" title="c7">q</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">r</font> = <font color="Maroon" title="c5">b1</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b2</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>;<br/>
<div><span class="lab"><font color="Green" title="E27">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b2</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font> )</span><br/><span class="kw">assume </span><a NAME="E1:31_1"/>
<font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b2</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">b</font> = <font color="Maroon" title="c9">u</font> as   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D12" title="PRE_POLY:def.12">PRE_POLY:def 12</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:31_1_1_1"/>
( <font color="Maroon" title="c5">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c6">b2</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Maroon" title="c10">b</font> or  not <font color="Maroon" title="c5">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c6">b2</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Maroon" title="c10">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:31_1_1_1_1"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
<font color="Maroon" title="c5">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c6">b2</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Maroon" title="c10">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b2</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c10">b</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">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c10">b</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">b3</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font><span class="kw"> such that </span><br/><a NAME="E3:31_1_1_1_1"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c11">b3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="termord.html#T1" title="TERMORD:th.1">TERMORD:1</a></span>;<br/><a NAME="E4:31_1_1_1_1"/><span class="lab"><font color="Green" title="E30">A4</font></span>: 
<font color="Maroon" title="c5">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b2</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c11">b3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_1_1_1_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="pre_poly.html#T35" title="PRE_POLY:th.35">PRE_POLY:35</a></span>;<br/><a NAME="E5:31_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">b2</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c11">b3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c5">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#T48" title="PRE_POLY:th.48">PRE_POLY:48</a></span>;<br/><a NAME="E6:31_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A5</font></span>: 
<font color="Maroon" title="c6">b2</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Maroon" title="c10">b</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c5">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="termord.html#T1" title="TERMORD:th.1">TERMORD:1</a></span>;<br/><a NAME="E7:31_1_1_1_1"/>
<font color="Maroon" title="c5">b1</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Maroon" title="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:31_1_1_1_1"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="termord.html#T1" title="TERMORD:th.1">TERMORD:1</a></span>;<br/><a NAME="E8:31_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b2</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b2</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c5">b1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D1" target="_self" title="POLYRED:def.1">Def1</a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b2</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c10">b</font> = 
<font color="Maroon" title="c4">p</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c5">b1</font></span>)</span> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c6">b2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:31_1_1_1_1"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="ref" href="polyred.html#D1" target="_self" title="POLYRED:def.1">Def1</a></span>
<br/>.= 
<font color="Maroon" title="c4">p</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c6">b2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#T36" title="PRE_POLY:th.36">PRE_POLY:36</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1_1_1_1"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="polyred.html#D1" target="_self" title="POLYRED:def.1">Def1</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:31_1_1_1_2"/><span class="lab"><font color="Green" title="E28">A6</font></span>: 
not <font color="Maroon" title="c5">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c6">b2</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Maroon" title="c10">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b2</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c10">b</font></span><br/><div class="add"><a NAME="E2:31_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A7</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D1" target="_self" title="POLYRED:def.1">Def1</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:31_1_1_1_2_1_1"/>
( <font color="Maroon" title="c5">b1</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Maroon" title="c10">b</font> or  not <font color="Maroon" title="c5">b1</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Maroon" title="c10">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:31_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E30">A8</font></span>: 
<font color="Maroon" title="c5">b1</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Maroon" title="c10">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b2</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c10">b</font></span><br/><div class="add"><div><span class="lab"><font color="Green" title="E31">A9</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_2_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:31_1_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c6">b2</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Maroon" title="c10">b</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c5">b1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:31_1_1_1_2_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c5">b1</font></span>)</span> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c5">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#T47" title="PRE_POLY:th.47">PRE_POLY:47</a></span>;<br/><a NAME="E3:31_1_1_1_2_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c5">b1</font></span>)</span> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c5">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E30">A8</font></span></a>, <a class="ref" href="pre_poly.html#T47" title="PRE_POLY:th.47">PRE_POLY:47</a></span>;<br/><a NAME="E4:31_1_1_1_2_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c5">b1</font></span>)</span> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b2</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c5">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#T35" title="PRE_POLY:th.35">PRE_POLY:35</a></span>;<br/><span class="kw">hence </span><a NAME="E5:31_1_1_1_2_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1_1_1_2"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="ref" href="termord.html#T1" title="TERMORD:th.1">TERMORD:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E3:31_1_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b2</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b2</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c5">b1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E30">A8</font></span></a>, <a class="ref" href="polyred.html#D1" target="_self" title="POLYRED:def.1">Def1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:31_1_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b2</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31_1_1_1_2"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="txt" href="#E2:31_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E31">A9</font></span></a>, <a class="ref" href="polyred.html#D1" target="_self" title="POLYRED:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:31_1_1_1_2_1_1_2"/>
not <font color="Maroon" title="c5">b1</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Maroon" title="c10">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b2</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c10">b</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:31_1_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b2</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31_1_1_1_2"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="ref" href="polyred.html#D1" target="_self" title="POLYRED:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:31_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b2</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c10">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:31_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b2</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font></span>)</span> = 
 <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
<br/>.= 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b2</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:31"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">b1</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b2</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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