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<span class="kw">let </span><font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c2">o2</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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">L</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  non  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>   <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>   <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a>  holds   <a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Olive" title="b1">L</font></span>)</span>, <a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Olive" title="b1">L</font> <a href="polynom6.html#NR1" title="POLYNOM6:NR.1">are_isomorphic</a> </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>  non  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>   <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>   <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span>, <a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font> <a href="polynom6.html#NR1" title="POLYNOM6:NR.1">are_isomorphic</a> </span><br/>



<span class="kw">set </span><font color="Maroon" title="c4">P2</font> =  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">P1</font> =  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>for <font color="Olive" title="b1">P</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P</font> holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Olive" title="b1">P</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">1P1</font> =  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">1P2</font> =  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font></span>)</span> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/>
<a NAME="E3:21"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span>  ex <font color="Olive" title="b2">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">u</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Olive" title="b1">u</font>]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">Q</font> = <font color="Maroon" title="c8">x</font> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">u</font> =  <a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c9">Q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c10">u</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c10">u</font>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c11">P</font> be   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span>; <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="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">P</font> implies <font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c11">P</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:21_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">P</font>
 ; <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="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c11">P</font></span><br/>

<span class="kw">hence </span><a NAME="E4:21_1"/>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c11">P</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>
<span class="kw">consider </span><font color="Maroon" title="c8">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</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#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span>,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#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E5:21"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E16">A1</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c9">f</font> = <font color="Maroon" title="c8">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font></span>)</span> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c9">f</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="quofield.html#D26" title="QUOFIELD:def.26">QUOFIELD:def 26</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">f</font> is  <a href="quofield.html#V4" title="QUOFIELD:attr.4">RingIsomorphism</a> </span><br/>

<span class="kw">thus </span><a NAME="E7:21"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
<font color="Maroon" title="c9">f</font> is  <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a> 
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="quofield.html#D21" title="QUOFIELD:def.21">QUOFIELD:def 21</a>,<a class="ref" href="quofield.html#D23" title="QUOFIELD:def.23">QUOFIELD:def 23</a>,<a class="ref" href="quofield.html#D24" title="QUOFIELD:def.24">QUOFIELD:def 24</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">f</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  &amp; <font color="Maroon" title="c9">f</font> is  <a href="group_1.html#V6" title="GROUP_1:attr.6">unity-preserving</a>  &amp; <font color="Maroon" title="c9">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c9">f</font> is  <a href="quofield.html#V2" title="QUOFIELD:attr.2">RingEpimorphism</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="grcat_1.html#D13" title="GRCAT_1:def.13">GRCAT_1:def 13</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">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</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="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c12">x9</font> = <font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c13">y9</font> = <font color="Maroon" title="c11">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">p</font> = <font color="Maroon" title="c12">x9</font>, <font color="Maroon" title="c15">q</font> = <font color="Maroon" title="c13">y9</font> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">fp</font> = <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c17">fq</font> = <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y</font>, <font color="Maroon" title="c18">fpq</font> = <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">y</font></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font></span>)</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">fp</font> = <font color="Maroon" title="c16">fp</font>, <font color="Maroon" title="c20">fq</font> = <font color="Maroon" title="c17">fq</font>, <font color="Maroon" title="c21">fpq</font> = <font color="Maroon" title="c18">fpq</font> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/>
<a NAME="E5:21_2"/>
for <font color="Olive" title="b1">x</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font> holds  <font color="Maroon" title="c21">fpq</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">fp</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">fq</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b1">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c22">b</font> be   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font>; <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="c21">fpq</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c22">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">fp</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c22">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">fq</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c22">b</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c23">b9</font> = <font color="Maroon" title="c22">b</font> as    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span> <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/>
<span class="kw">consider </span><font color="Maroon" title="c24">b1</font> being    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c25">b2</font> being    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font>, <font color="Maroon" title="c26">Q1</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E3:21_2_1"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
<font color="Maroon" title="c26">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">p</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c24">b1</font>
 <span class="kw">and </span><br/><a NAME="E4:21_2_1"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<font color="Maroon" title="c23">b9</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c24">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c25">b2</font>
 <span class="kw">and </span><br/><a NAME="E5:21_2_1"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
<span class="p1">(<span class="default"><a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c14">p</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c23">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">Q1</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c25">b2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom6.html#D2" target="_self" title="POLYNOM6:def.2">Def2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c27">b19</font> being    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c28">b29</font> being    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font>, <font color="Maroon" title="c29">Q19</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E7:21_2_1"/><span class="lab"><font color="Green" title="E22">A7</font></span>: 
<font color="Maroon" title="c29">Q19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c27">b19</font>
 <span class="kw">and </span><br/><a NAME="E8:21_2_1"/><span class="lab"><font color="Green" title="E23">A8</font></span>: 
<font color="Maroon" title="c23">b9</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c27">b19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c28">b29</font>
 <span class="kw">and </span><br/><a NAME="E9:21_2_1"/><span class="lab"><font color="Green" title="E24">A9</font></span>: 
<span class="p1">(<span class="default"><a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c15">q</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c23">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">Q19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c28">b29</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom6.html#D2" target="_self" title="POLYNOM6:def.2">Def2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c30">b199</font> being    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c31">b299</font> being    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font>, <font color="Maroon" title="c32">Q199</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E11:21_2_1"/><span class="lab"><font color="Green" title="E25">A10</font></span>: 
<font color="Maroon" title="c32">Q199</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">+</a> <font color="Maroon" title="c15">q</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c30">b199</font>
 <span class="kw">and </span><br/><a NAME="E12:21_2_1"/><span class="lab"><font color="Green" title="E26">A11</font></span>: 
<font color="Maroon" title="c23">b9</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c30">b199</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c31">b299</font>
 <span class="kw">and </span><br/><a NAME="E13:21_2_1"/><span class="lab"><font color="Green" title="E27">A12</font></span>: 
<span class="p1">(<span class="default"><a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">+</a> <font color="Maroon" title="c15">q</font></span>)</span></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c23">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">Q199</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c31">b299</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom6.html#D2" target="_self" title="POLYNOM6:def.2">Def2</a></span>;<br/>
<a NAME="E14:21_2_1"/><span class="lab"><font color="Green" title="E28">A13</font></span>: 
<font color="Maroon" title="c27">b19</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c30">b199</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_2_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E12:21_2_1"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="polynom6.html#T7" target="_self" title="POLYNOM6:th.7">Th7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c33">b1</font> = <font color="Maroon" title="c24">b1</font> as   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">o1</font> ;<br/>
<a NAME="E16:21_2_1"/><span class="lab"><font color="Green" title="E29">A14</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">+</a> <font color="Maroon" title="c15">q</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c33">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c33">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">q</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c33">b1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D21" title="POLYNOM1:def.21">POLYNOM1:def 21</a></span>;<br/>
<a NAME="E17:21_2_1"/>
<font color="Maroon" title="c33">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c27">b19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_2_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E8:21_2_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="polynom6.html#T7" target="_self" title="POLYNOM6:th.7">Th7</a></span>;<br/>
<a NAME="E18:21_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c32">Q199</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c26">Q1</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">+</a> <font color="Maroon" title="c29">Q19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_2_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E7:21_2_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E11:21_2_1"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="txt" href="#E14:21_2_1"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E16:21_2_1"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/>
<a NAME="E19:21_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A15</font></span>: 
<font color="Maroon" title="c32">Q199</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c25">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">Q1</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">Q19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c25">b2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D21" title="POLYNOM1:def.21">POLYNOM1:def 21</a></span>;<br/>
<a NAME="E20:21_2_1"/><span class="lab"><font color="Green" title="E31">A16</font></span>: 
<font color="Maroon" title="c28">b29</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c31">b299</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_2_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E12:21_2_1"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="polynom6.html#T7" target="_self" title="POLYNOM6:th.7">Th7</a></span>;<br/>
<a NAME="E21:21_2_1"/><span class="lab"><font color="Green" title="E32">A17</font></span>: 
<font color="Maroon" title="c25">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c28">b29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_2_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E8:21_2_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="polynom6.html#T7" target="_self" title="POLYNOM6:th.7">Th7</a></span>;<br/>
<a NAME="E22:21_2_1"/>
<font color="Maroon" title="c10">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">p</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">+</a> <font color="Maroon" title="c15">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c21">fpq</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c22">b</font> = 
<span class="p1">(<span class="default"><a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">+</a> <font color="Maroon" title="c15">q</font></span>)</span></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c23">b9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21"><span class="lab"><font color="Green" title="E17">A2</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c14">p</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c23">b9</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">fq</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c22">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E5:21_2_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E9:21_2_1"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E13:21_2_1"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="txt" href="#E21:21_2_1"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E20:21_2_1"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="txt" href="#E19:21_2_1"><span class="lab"><font color="Green" title="E30">A15</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">fp</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c22">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">fq</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c22">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21"><span class="lab"><font color="Green" title="E17">A2</font></span></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>
<span class="kw">hence </span><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">y</font></span>)</span> = 
<font color="Maroon" title="c19">fp</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">+</a> <font color="Maroon" title="c20">fq</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D21" title="POLYNOM1:def.21">POLYNOM1:def 21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</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="hs2(this)" href="javascript:()" title="21_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span>; <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">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y</font></span>)</span></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c12">x9</font> = <font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c13">y9</font> = <font color="Maroon" title="c11">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">p</font> = <font color="Maroon" title="c12">x9</font>, <font color="Maroon" title="c15">q</font> = <font color="Maroon" title="c13">y9</font> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">fp</font> = <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c17">fq</font> = <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font></span>)</span> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">fp</font> = <font color="Maroon" title="c16">fp</font>, <font color="Maroon" title="c19">fq</font> = <font color="Maroon" title="c17">fq</font> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/><a NAME="E5:21_3"/>
<font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c15">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">fpq9</font> = <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c15">q</font></span>)</span> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/><a NAME="E7:21_3"/><span class="lab"><font color="Green" title="E19">A18</font></span>: 
for <font color="Olive" title="b1">b</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font>  ex <font color="Olive" title="b2">s</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font> st <br/>( <font color="Maroon" title="c20">fpq9</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Olive" title="b2">s</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Olive" title="b1">b</font></span>)</span> &amp; ( for <font color="Olive" title="b3">k</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>   st <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b2">s</font> holds <br/> ex <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">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b4">b1</font>,<font color="Olive" title="b5">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Olive" title="b2">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">fp</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b4">b1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">fq</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b5">b2</font></span>)</span> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c21">x</font> = <font color="Maroon" title="c14">p</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c15">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c22">c</font> be   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">s</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font> st <br/>( <font color="Maroon" title="c20">fpq9</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c22">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Olive" title="b1">s</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c22">c</font></span>)</span> &amp; ( for <font color="Olive" title="b2">k</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>   st <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">s</font> holds <br/> ex <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">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c22">c</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b3">b1</font>,<font color="Olive" title="b4">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Olive" title="b1">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">fp</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b3">b1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">fq</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b4">b2</font></span>)</span> ) ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c23">b</font> = <font color="Maroon" title="c22">c</font> as    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span> <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/>
<span class="kw">consider </span><font color="Maroon" title="c24">b1</font> being    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c25">b2</font> being    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font><span class="kw"> such that </span><br/><a NAME="E4:21_3_1"/><span class="lab"><font color="Green" title="E20">A19</font></span>: 
<font color="Maroon" title="c23">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c24">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c25">b2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom6.html#T6" target="_self" title="POLYNOM6:th.6">Th6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c26">b1</font> = <font color="Maroon" title="c24">b1</font> as   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">o1</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c27">r</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</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#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E7:21_3_1"/><span class="lab"><font color="Green" title="E21">A20</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c15">q</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c26">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c27">r</font>
 <span class="kw">and </span><br/><a NAME="E8:21_3_1"/><span class="lab"><font color="Green" title="E22">A21</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c26">b1</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E9:21_3_1"/><span class="lab"><font color="Green" title="E23">A22</font></span>: 
for <font color="Olive" title="b1">k</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>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c27">r</font> holds <br/> ex <font color="Olive" title="b2">a19</font>, <font color="Olive" title="b3">b19</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">o1</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c26">b1</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b2">a19</font>,<font color="Olive" title="b3">b19</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c27">r</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b2">a19</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">q</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b3">b19</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D26" title="POLYNOM1:def.26">POLYNOM1:def 26</a></span>;<br/>
<a NAME="E10:21_3_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c27">r</font> holds <br/><font color="Maroon" title="c27">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c28">x</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="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c27">r</font> implies <font color="Maroon" title="c27">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_3_1_1"/>
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c27">r</font>
 ; <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="c27">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a></span><br/>

<a NAME="E2:21_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A23</font></span>: 
<font color="Maroon" title="c27">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c27">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<a NAME="E3:21_3_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c27">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:21_3_1_1"/>
<font color="Maroon" title="c27">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_3_1_1"><span class="lab"><font color="Green" title="E24">A23</font></span></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</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>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c28">rFF</font> = <font color="Maroon" title="c27">r</font> as   <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#D6" title="FUNCOP_1:def.6">FUNCOP_1:def 6</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">(<span class="default"><font color="Maroon" title="c28">rFF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">b2</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c29">rFFb2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E13:21_3_1"/><span class="lab"><font color="Green" title="E24">A24</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c29">rFFb2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c27">r</font>
 <span class="kw">and </span><br/><a NAME="E14:21_3_1"/><span class="lab"><font color="Green" title="E25">A25</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c27">r</font> holds <br/><font color="Maroon" title="c29">rFFb2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></span>
<a NAME="E15:21_3_1"/>
 ex <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c27">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Olive" title="b1">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c30">rFFb2</font> = <font color="Maroon" title="c29">rFFb2</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21_3_1"><span class="lab"><font color="Green" title="E24">A24</font></span></a>, <a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></span>;<br/>
<a NAME="E17:21_3_1"/><span class="lab"><font color="Green" title="E26">A26</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c30">rFFb2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c31">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c31">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c30">rFFb2</font> or <font color="Maroon" title="c31">u</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 <font color="Maroon" title="c3">L</font> )</span><br/>

<a NAME="E1:21_3_1_2"/><span class="lab"><font color="Green" title="E27">A27</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c28">rFF</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:21_3_1_2"/>
<font color="Maroon" title="c31">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c30">rFFb2</font>
 ; <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="c31">u</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 <font color="Maroon" title="c3">L</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c32">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:21_3_1_2"/><span class="lab"><font color="Green" title="E28">A28</font></span>: 
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c30">rFFb2</font>
 <span class="kw">and </span><br/><a NAME="E5:21_3_1_2"/><span class="lab"><font color="Green" title="E29">A29</font></span>: 
<font color="Maroon" title="c31">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">rFFb2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E6:21_3_1_2"/>
<font color="Maroon" title="c28">rFF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c28">rFF</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21_3_1"><span class="lab"><font color="Green" title="E24">A24</font></span></a>, <a class="txt" href="#E4:21_3_1_2"><span class="lab"><font color="Green" title="E28">A28</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<a NAME="E7:21_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A30</font></span>: 
<font color="Maroon" title="c28">rFF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_3_1_2"><span class="lab"><font color="Green" title="E27">A27</font></span></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/>
<a NAME="E8:21_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A31</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">rFF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<a NAME="E9:21_3_1_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">rFF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_3_1_2"><span class="lab"><font color="Green" title="E30">A30</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E10:21_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A32</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c28">rFF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">rFF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<a NAME="E11:21_3_1_2"/>
<font color="Maroon" title="c30">rFFb2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">rFF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">b2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21_3_1"><span class="lab"><font color="Green" title="E24">A24</font></span></a>, <a class="txt" href="#E14:21_3_1"><span class="lab"><font color="Green" title="E25">A25</font></span></a>, <a class="txt" href="#E4:21_3_1_2"><span class="lab"><font color="Green" title="E28">A28</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:21_3_1_2"/>
<font color="Maroon" title="c31">u</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 <font color="Maroon" title="c3">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21_3_1_2"><span class="lab"><font color="Green" title="E29">A29</font></span></a>, <a class="txt" href="#E8:21_3_1_2"><span class="lab"><font color="Green" title="E31">A31</font></span></a>, <a class="txt" href="#E10:21_3_1_2"><span class="lab"><font color="Green" title="E32">A32</font></span></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>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">a19</font>, <font color="Olive" title="b2">b19</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">o1</font> ex <font color="Olive" title="b3">Fk</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font> ex <font color="Olive" title="b4">pa19</font>, <font color="Olive" title="b5">qb19</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font> st <br/>( <font color="Olive" title="b4">pa19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">p</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b1">a19</font> &amp; <font color="Olive" title="b5">qb19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b2">b19</font> &amp; <font color="Olive" title="b3">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c26">b1</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b1">a19</font>,<font color="Olive" title="b2">b19</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> &amp; ( for <font color="Olive" title="b6">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b3">Fk</font> holds <br/> ex <font color="Olive" title="b7">a199</font>, <font color="Olive" title="b8">b199</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b7">a199</font>,<font color="Olive" title="b8">b199</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Olive" title="b3">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">pa19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b7">a199</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">qb19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b8">b199</font></span>)</span> ) ) );<br/>
<a NAME="E18:21_3_1"/><span class="lab"><font color="Green" title="E27">A33</font></span>: 
for <font color="Olive" title="b1">k</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>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">r</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c31">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c31">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">r</font></span>)</span> implies  ex <font color="Olive" title="b1">x</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c31">k</font>,<font color="Olive" title="b1">x</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_3_1_3"/>
<font color="Maroon" title="c31">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">r</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c31">k</font>,<font color="Olive" title="b1">x</font>]</span><br/>

<a NAME="E2:21_3_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c31">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c26">b1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_3_1"><span class="lab"><font color="Green" title="E22">A21</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c32">a19</font>, <font color="Maroon" title="c33">b19</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">o1</font><span class="kw"> such that </span><br/><a NAME="E4:21_3_1_3"/><span class="lab"><font color="Green" title="E28">A34</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c26">b1</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c31">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c32">a19</font>,<font color="Maroon" title="c33">b19</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E5:21_3_1_3"/>
<font color="Maroon" title="c26">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c32">a19</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c33">b19</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#T68" title="PRE_POLY:th.68">PRE_POLY:68</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c34">pa199</font> = <font color="Maroon" title="c14">p</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c32">a19</font>, <font color="Maroon" title="c35">qb199</font> = <font color="Maroon" title="c15">q</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c33">b19</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c36">pa19</font> = <font color="Maroon" title="c34">pa199</font>, <font color="Maroon" title="c37">qb19</font> = <font color="Maroon" title="c35">qb199</font> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">a199</font>, <font color="Olive" title="b2">b199</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b1">a199</font>,<font color="Olive" title="b2">b199</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">pa19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b1">a199</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">qb19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b2">b199</font></span>)</span> );<br/>
<a NAME="E8:21_3_1_3"/><span class="lab"><font color="Green" title="E29">A35</font></span>: 
for <font color="Olive" title="b1">k</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>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span></span>)</span> holds <br/> ex <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c38">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c38">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span></span>)</span> implies  ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c38">k</font>,<font color="Olive" title="b1">x</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_3_1_3_1"/>
<font color="Maroon" title="c38">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c38">k</font>,<font color="Olive" title="b1">x</font>]</span><br/>

<a NAME="E2:21_3_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c38">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c39">a199</font>, <font color="Maroon" title="c40">b199</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">o2</font><span class="kw"> such that </span><br/><a NAME="E4:21_3_1_3_1"/><span class="lab"><font color="Green" title="E30">A36</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c38">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c39">a199</font>,<font color="Maroon" title="c40">b199</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E5:21_3_1_3_1"/>
<font color="Maroon" title="c25">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c39">a199</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c40">b199</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#T68" title="PRE_POLY:th.68">PRE_POLY:68</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c41">x</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c36">pa19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c39">a199</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">qb19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c40">b199</font></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c41">x</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c38">k</font>,<font color="Maroon" title="c41">x</font>]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c39">a199</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">b199</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c38">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c39">a199</font>,<font color="Olive" title="b1">b199</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c41">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">pa19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c39">a199</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">qb19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b1">b199</font></span>)</span> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c40">b199</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"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c38">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c39">a199</font>,<font color="Maroon" title="c40">b199</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c41">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">pa19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c39">a199</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">qb19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c40">b199</font></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E7:21_3_1_3_1"/>
( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c38">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c39">a199</font>,<font color="Maroon" title="c40">b199</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c41">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">pa19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c39">a199</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">qb19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c40">b199</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_3_1_3_1"><span class="lab"><font color="Green" title="E30">A36</font></span></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>
<span class="kw">consider </span><font color="Maroon" title="c38">Fk</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E10:21_3_1_3"/><span class="lab"><font color="Green" title="E30">A37</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c38">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E11:21_3_1_3"/><span class="lab"><font color="Green" title="E31">A38</font></span>: 
for <font color="Olive" title="b1">k</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>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c38">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="recdef_1.html#S17" title="RECDEF_1:sch.17">RECDEF_1:sch 17</a>(<a class="txt" href="#E8:21_3_1_3"><span class="lab"><font color="Green" title="E29">A35</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c39">x</font> = <font color="Maroon" title="c38">Fk</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c39">x</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c31">k</font>,<font color="Maroon" title="c39">x</font>]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c32">a19</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">b19</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">o1</font> ex <font color="Olive" title="b2">Fk</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font> ex <font color="Olive" title="b3">pa19</font>, <font color="Olive" title="b4">qb19</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font> st <br/>( <font color="Olive" title="b3">pa19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">p</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c32">a19</font> &amp; <font color="Olive" title="b4">qb19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b1">b19</font> &amp; <font color="Olive" title="b2">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">x</font> &amp; <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c26">b1</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c31">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c32">a19</font>,<font color="Olive" title="b1">b19</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> &amp; ( for <font color="Olive" title="b5">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b5">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b2">Fk</font> holds <br/> ex <font color="Olive" title="b6">a199</font>, <font color="Olive" title="b7">b199</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b6">a199</font>,<font color="Olive" title="b7">b199</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Olive" title="b2">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">pa19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b6">a199</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">qb19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b7">b199</font></span>)</span> ) ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c33">b19</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">Fk</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font> ex <font color="Olive" title="b2">pa19</font>, <font color="Olive" title="b3">qb19</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font> st <br/>( <font color="Olive" title="b2">pa19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">p</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c32">a19</font> &amp; <font color="Olive" title="b3">qb19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c33">b19</font> &amp; <font color="Olive" title="b1">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">x</font> &amp; <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c26">b1</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c31">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c32">a19</font>,<font color="Maroon" title="c33">b19</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> &amp; ( for <font color="Olive" title="b4">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">Fk</font> holds <br/> ex <font color="Olive" title="b5">a199</font>, <font color="Olive" title="b6">b199</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b5">a199</font>,<font color="Olive" title="b6">b199</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Olive" title="b1">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">pa19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b5">a199</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">qb19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b6">b199</font></span>)</span> ) ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c38">Fk</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">pa19</font>, <font color="Olive" title="b2">qb19</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font> st <br/>( <font color="Olive" title="b1">pa19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">p</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c32">a19</font> &amp; <font color="Olive" title="b2">qb19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c33">b19</font> &amp; <font color="Maroon" title="c38">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">x</font> &amp; <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c26">b1</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c31">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c32">a19</font>,<font color="Maroon" title="c33">b19</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> &amp; ( for <font color="Olive" title="b3">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c38">Fk</font> holds <br/> ex <font color="Olive" title="b4">a199</font>, <font color="Olive" title="b5">b199</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b4">a199</font>,<font color="Olive" title="b5">b199</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c38">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">pa19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b4">a199</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">qb19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b5">b199</font></span>)</span> ) ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c36">pa19</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">qb19</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font> st <br/>( <font color="Maroon" title="c36">pa19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">p</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c32">a19</font> &amp; <font color="Olive" title="b1">qb19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c33">b19</font> &amp; <font color="Maroon" title="c38">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">x</font> &amp; <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c26">b1</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c31">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c32">a19</font>,<font color="Maroon" title="c33">b19</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> &amp; ( for <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c38">Fk</font> holds <br/> ex <font color="Olive" title="b3">a199</font>, <font color="Olive" title="b4">b199</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b3">a199</font>,<font color="Olive" title="b4">b199</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c38">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">pa19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b3">a199</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">qb19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b4">b199</font></span>)</span> ) ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c37">qb19</font>
; <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="c36">pa19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">p</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c32">a19</font> &amp; <font color="Maroon" title="c37">qb19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c33">b19</font> &amp; <font color="Maroon" title="c38">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">x</font> &amp; <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c26">b1</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c31">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c32">a19</font>,<font color="Maroon" title="c33">b19</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> &amp; ( for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c38">Fk</font> holds <br/> ex <font color="Olive" title="b2">a199</font>, <font color="Olive" title="b3">b199</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b2">a199</font>,<font color="Olive" title="b3">b199</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c38">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">pa19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b2">a199</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">qb19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b3">b199</font></span>)</span> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E13:21_3_1_3"/>
( <font color="Maroon" title="c36">pa19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">p</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c32">a19</font> &amp; <font color="Maroon" title="c37">qb19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c33">b19</font> &amp; <font color="Maroon" title="c38">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">x</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"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c26">b1</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c31">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c32">a19</font>,<font color="Maroon" title="c33">b19</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> &amp; ( for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c38">Fk</font> holds <br/> ex <font color="Olive" title="b2">a199</font>, <font color="Olive" title="b3">b199</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b2">a199</font>,<font color="Olive" title="b3">b199</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c38">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">pa19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b2">a199</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">qb19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b3">b199</font></span>)</span> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E14:21_3_1_3"/>
<span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c26">b1</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c31">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c32">a19</font>,<font color="Maroon" title="c33">b19</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_3_1_3"><span class="lab"><font color="Green" title="E28">A34</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> &amp; ( for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c38">Fk</font> holds <br/> ex <font color="Olive" title="b2">a199</font>, <font color="Olive" title="b3">b199</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b2">a199</font>,<font color="Olive" title="b3">b199</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c38">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">pa19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b2">a199</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">qb19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b3">b199</font></span>)</span> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E15:21_3_1_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:21_3_1_3"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>; <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">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c38">Fk</font> holds <br/> ex <font color="Olive" title="b2">a199</font>, <font color="Olive" title="b3">b199</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b2">a199</font>,<font color="Olive" title="b3">b199</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c38">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">pa19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b2">a199</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">qb19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b3">b199</font></span>)</span> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c40">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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="c40">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c38">Fk</font> implies  ex <font color="Olive" title="b1">a199</font>, <font color="Olive" title="b2">b199</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c40">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b1">a199</font>,<font color="Olive" title="b2">b199</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c38">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c40">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">pa19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b1">a199</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">qb19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b2">b199</font></span>)</span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E16:21_3_1_3"/>
<font color="Maroon" title="c40">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c38">Fk</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a199</font>, <font color="Olive" title="b2">b199</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c40">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b1">a199</font>,<font color="Olive" title="b2">b199</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c38">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c40">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">pa19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b1">a199</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">qb19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b2">b199</font></span>)</span> )</span><br/>

<span class="kw">hence </span><a NAME="E17:21_3_1_3"/>
 ex <font color="Olive" title="b1">a199</font>, <font color="Olive" title="b2">b199</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c40">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b1">a199</font>,<font color="Olive" title="b2">b199</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c38">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c40">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">pa19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b1">a199</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">qb19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b2">b199</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:21_3_1_3"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E11:21_3_1_3"><span class="lab"><font color="Green" title="E31">A38</font></span></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>
<span class="kw">consider </span><font color="Maroon" title="c31">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> <span class="kw"> such that </span><br/><a NAME="E20:21_3_1"/><span class="lab"><font color="Green" title="E28">A39</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c31">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">r</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E21:21_3_1"/><span class="lab"><font color="Green" title="E29">A40</font></span>: 
for <font color="Olive" title="b1">k</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>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">r</font></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c31">F</font> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">k</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="recdef_1.html#S17" title="RECDEF_1:sch.17">RECDEF_1:sch 17</a>(<a class="txt" href="#E18:21_3_1"><span class="lab"><font color="Green" title="E27">A33</font></span></a>);<br/></span>
<span class="kw">take </span><font color="Maroon" title="c32">s</font> =  <a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c31">F</font>; <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="c20">fpq9</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c22">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c32">s</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c22">c</font></span>)</span> &amp; ( for <font color="Olive" title="b1">k</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>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c32">s</font> holds <br/> ex <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">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c22">c</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b2">b1</font>,<font color="Olive" title="b3">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c32">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">fp</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b2">b1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">fq</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b3">b2</font></span>)</span> ) ) )</span><br/>

<a NAME="E22:21_3_1"/><span class="lab"><font color="Green" title="E30">A41</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c31">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c31">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#D8" title="MATRLIN:def.8">MATRLIN:def 8</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c33">rFFb2</font> = <font color="Maroon" title="c30">rFFb2</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:21_3_1"><span class="lab"><font color="Green" title="E26">A26</font></span></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<a NAME="E24:21_3_1"/><span class="lab"><font color="Green" title="E31">A42</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c15">q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21"><span class="lab"><font color="Green" title="E17">A2</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E32">A43</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c33">rFFb2</font> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c31">F</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:21_3_1"><span class="lab"><font color="Green" title="E28">A39</font></span></a>, <a class="txt" href="#E13:21_3_1"><span class="lab"><font color="Green" title="E24">A24</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c31">F</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:21_3_1"><span class="lab"><font color="Green" title="E30">A41</font></span></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>
;<br/>
<a NAME="E26:21_3_1"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c33">rFFb2</font> holds <br/><font color="Maroon" title="c33">rFFb2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c31">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c34">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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="c34">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c33">rFFb2</font> implies <font color="Maroon" title="c33">rFFb2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c31">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">k</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_3_1_5"/><span class="lab"><font color="Green" title="E33">A44</font></span>: 
<font color="Maroon" title="c34">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c33">rFFb2</font>
 ; <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="c33">rFFb2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c31">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">k</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c35">c1</font>, <font color="Maroon" title="c36">d1</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">o1</font><span class="kw"> such that </span><br/><a NAME="E3:21_3_1_5"/><span class="lab"><font color="Green" title="E34">A45</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c26">b1</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c34">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c35">c1</font>,<font color="Maroon" title="c36">d1</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E4:21_3_1_5"/><span class="lab"><font color="Green" title="E35">A46</font></span>: 
<font color="Maroon" title="c27">r</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c34">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c35">c1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">q</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c36">d1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:21_3_1"><span class="lab"><font color="Green" title="E23">A22</font></span></a>, <a class="txt" href="#E13:21_3_1"><span class="lab"><font color="Green" title="E24">A24</font></span></a>, <a class="txt" href="#E1:21_3_1_5"><span class="lab"><font color="Green" title="E33">A44</font></span></a></span>;<br/>
<a NAME="E5:21_3_1_5"/>
<font color="Maroon" title="c34">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">r</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21_3_1"><span class="lab"><font color="Green" title="E24">A24</font></span></a>, <a class="txt" href="#E1:21_3_1_5"><span class="lab"><font color="Green" title="E33">A44</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c37">a19</font>, <font color="Maroon" title="c38">b19</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c39">Fk</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font>, <font color="Maroon" title="c40">pa19</font>, <font color="Maroon" title="c41">qb19</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E7:21_3_1_5"/><span class="lab"><font color="Green" title="E36">A47</font></span>: 
( <font color="Maroon" title="c40">pa19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">p</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c37">a19</font> &amp; <font color="Maroon" title="c41">qb19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c38">b19</font> )
 <span class="kw">and </span><br/><a NAME="E8:21_3_1_5"/><span class="lab"><font color="Green" title="E37">A48</font></span>: 
<font color="Maroon" title="c39">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">F</font> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c34">k</font>
 <span class="kw">and </span><br/><a NAME="E9:21_3_1_5"/><span class="lab"><font color="Green" title="E38">A49</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c26">b1</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c34">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c37">a19</font>,<font color="Maroon" title="c38">b19</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E10:21_3_1_5"/><span class="lab"><font color="Green" title="E39">A50</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E11:21_3_1_5"/><span class="lab"><font color="Green" title="E40">A51</font></span>: 
for <font color="Olive" title="b1">ki</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">ki</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c39">Fk</font> holds <br/> ex <font color="Olive" title="b2">a199</font>, <font color="Olive" title="b3">b199</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">ki</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b2">a199</font>,<font color="Olive" title="b3">b199</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c39">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">ki</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c40">pa19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b2">a199</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">qb19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b3">b199</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:21_3_1"><span class="lab"><font color="Green" title="E29">A40</font></span></a></span>;<br/>
<a NAME="E12:21_3_1_5"/><span class="lab"><font color="Green" title="E41">A52</font></span>: 
( <font color="Maroon" title="c35">c1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c37">a19</font> &amp; <font color="Maroon" title="c36">d1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c38">b19</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_3_1_5"><span class="lab"><font color="Green" title="E34">A45</font></span></a>, <a class="txt" href="#E9:21_3_1_5"><span class="lab"><font color="Green" title="E38">A49</font></span></a>, <a class="ref" href="finseq_1.html#T98" title="FINSEQ_1:th.98">FINSEQ_1:98</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c42">s1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E14:21_3_1_5"/><span class="lab"><font color="Green" title="E42">A53</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c40">pa19</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c41">qb19</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c25">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c42">s1</font>
 <span class="kw">and </span><br/><a NAME="E15:21_3_1_5"/><span class="lab"><font color="Green" title="E43">A54</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E16:21_3_1_5"/><span class="lab"><font color="Green" title="E44">A55</font></span>: 
for <font color="Olive" title="b1">ki</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>   st <font color="Olive" title="b1">ki</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c42">s1</font> holds <br/> ex <font color="Olive" title="b2">x1</font>, <font color="Olive" title="b3">y2</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">ki</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b2">x1</font>,<font color="Olive" title="b3">y2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c42">s1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">ki</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c40">pa19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b2">x1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">qb19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b3">y2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D26" title="POLYNOM1:def.26">POLYNOM1:def 26</a></span>;<br/>
<a NAME="E17:21_3_1_5"/><span class="lab"><font color="Green" title="E45">A56</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c42">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">s1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_1_5_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c43">ki</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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="c43">ki</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c42">s1</font> implies <font color="Maroon" title="c42">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">ki</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">Fk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">ki</font> )</span><br/><span class="kw">assume </span><a NAME="E1:21_3_1_5_1"/><span class="lab"><font color="Green" title="E46">A57</font></span>: 
<font color="Maroon" title="c43">ki</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c42">s1</font>
 ; <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="c42">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">ki</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">Fk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">ki</font></span><br/><a NAME="E2:21_3_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A58</font></span>: 
<font color="Maroon" title="c42">s1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c43">ki</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">ki</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E3:21_3_1_5_1"/><span class="lab"><font color="Green" title="E48">A59</font></span>: 
<font color="Maroon" title="c43">ki</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c39">Fk</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:21_3_1_5"><span class="lab"><font color="Green" title="E39">A50</font></span></a>, <a class="txt" href="#E15:21_3_1_5"><span class="lab"><font color="Green" title="E43">A54</font></span></a>, <a class="txt" href="#E17:21_3_1_5"><span class="lab"><font color="Green" title="E45">A56</font></span></a>, <a class="txt" href="#E1:21_3_1_5_1"><span class="lab"><font color="Green" title="E46">A57</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c44">a199</font>, <font color="Maroon" title="c45">b199</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">o2</font><span class="kw"> such that </span><br/><a NAME="E5:21_3_1_5_1"/><span class="lab"><font color="Green" title="E49">A60</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c43">ki</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c44">a199</font>,<font color="Maroon" title="c45">b199</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E6:21_3_1_5_1"/><span class="lab"><font color="Green" title="E50">A61</font></span>: 
<font color="Maroon" title="c39">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c43">ki</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c40">pa19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c44">a199</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">qb19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c45">b199</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:21_3_1_5"><span class="lab"><font color="Green" title="E40">A51</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c46">x1</font>, <font color="Maroon" title="c47">y2</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">o2</font><span class="kw"> such that </span><br/><a NAME="E8:21_3_1_5_1"/><span class="lab"><font color="Green" title="E51">A62</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c43">ki</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c46">x1</font>,<font color="Maroon" title="c47">y2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E9:21_3_1_5_1"/><span class="lab"><font color="Green" title="E52">A63</font></span>: 
<font color="Maroon" title="c42">s1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c43">ki</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c40">pa19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c46">x1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">qb19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c47">y2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:21_3_1_5"><span class="lab"><font color="Green" title="E44">A55</font></span></a>, <a class="txt" href="#E1:21_3_1_5_1"><span class="lab"><font color="Green" title="E46">A57</font></span></a></span>;<br/><a NAME="E10:21_3_1_5_1"/>
( <font color="Maroon" title="c46">x1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c44">a199</font> &amp; <font color="Maroon" title="c47">y2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c45">b199</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_3_1_5_1"><span class="lab"><font color="Green" title="E51">A62</font></span></a>, <a class="txt" href="#E5:21_3_1_5_1"><span class="lab"><font color="Green" title="E49">A60</font></span></a>, <a class="ref" href="finseq_1.html#T98" title="FINSEQ_1:th.98">FINSEQ_1:98</a></span>;<br/><span class="kw">hence </span><a NAME="E11:21_3_1_5_1"/>
<font color="Maroon" title="c42">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">ki</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">Fk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">ki</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:21_3_1_5_1"><span class="lab"><font color="Green" title="E52">A63</font></span></a>, <a class="txt" href="#E3:21_3_1_5_1"><span class="lab"><font color="Green" title="E48">A59</font></span></a>, <a class="txt" href="#E6:21_3_1_5_1"><span class="lab"><font color="Green" title="E50">A61</font></span></a>, <a class="txt" href="#E2:21_3_1_5_1"><span class="lab"><font color="Green" title="E47">A58</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</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="E19:21_3_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A64</font></span>: 
<font color="Maroon" title="c42">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">Fk</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:21_3_1_5"><span class="lab"><font color="Green" title="E39">A50</font></span></a>, <a class="txt" href="#E15:21_3_1_5"><span class="lab"><font color="Green" title="E43">A54</font></span></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></span>;<br/>
<span class="lab"><font color="Green" title="E47">A65</font></span>: <font color="Maroon" title="c28">rFF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">k</font> = 
<font color="Maroon" title="c27">r</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c34">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21_3_1"><span class="lab"><font color="Green" title="E24">A24</font></span></a>, <a class="txt" href="#E1:21_3_1_5"><span class="lab"><font color="Green" title="E33">A44</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c40">pa19</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c41">qb19</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_3_1_5"><span class="lab"><font color="Green" title="E35">A46</font></span></a>, <a class="txt" href="#E7:21_3_1_5"><span class="lab"><font color="Green" title="E36">A47</font></span></a>, <a class="txt" href="#E12:21_3_1_5"><span class="lab"><font color="Green" title="E41">A52</font></span></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>
;<br/>
<span class="kw">thus </span><font color="Maroon" title="c33">rFFb2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c28">rFF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">b2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21_3_1"><span class="lab"><font color="Green" title="E24">A24</font></span></a>, <a class="txt" href="#E14:21_3_1"><span class="lab"><font color="Green" title="E25">A25</font></span></a>, <a class="txt" href="#E1:21_3_1_5"><span class="lab"><font color="Green" title="E33">A44</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c31">F</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c34">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:21_3_1"><span class="lab"><font color="Green" title="E32">A43</font></span></a>, <a class="txt" href="#E1:21_3_1_5"><span class="lab"><font color="Green" title="E33">A44</font></span></a>, <a class="txt" href="#E8:21_3_1_5"><span class="lab"><font color="Green" title="E37">A48</font></span></a>, <a class="txt" href="#E20:21_3_1_5"><span class="lab"><font color="Green" title="E47">A65</font></span></a>, <a class="txt" href="#E14:21_3_1_5"><span class="lab"><font color="Green" title="E42">A53</font></span></a>, <a class="txt" href="#E19:21_3_1_5"><span class="lab"><font color="Green" title="E46">A64</font></span></a>, <a class="ref" href="matrlin.html#D8" title="MATRLIN:def.8">MATRLIN:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c31">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:21_3_1"><span class="lab"><font color="Green" title="E32">A43</font></span></a>, <a class="txt" href="#E1:21_3_1_5"><span class="lab"><font color="Green" title="E33">A44</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</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="E27:21_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A66</font></span>: 
<font color="Maroon" title="c33">rFFb2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c31">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:21_3_1"><span class="lab"><font color="Green" title="E32">A43</font></span></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c34">Sr</font> =  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c27">r</font> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c35">gg</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E30:21_3_1"/><span class="lab"><font color="Green" title="E34">A67</font></span>: 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c27">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">gg</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">r</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E31:21_3_1"/><span class="lab"><font color="Green" title="E35">A68</font></span>: 
<font color="Maroon" title="c35">gg</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E32:21_3_1"/><span class="lab"><font color="Green" title="E36">A69</font></span>: 
for <font color="Olive" title="b1">j</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="b2">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span>  st <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">r</font> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> holds <br/><font color="Maroon" title="c35">gg</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c35">gg</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D15" title="RLVECT_1:def.15">RLVECT_1:def 15</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>for <font color="Olive" title="b1">pp</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font>  st $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">r</font> &amp; <font color="Olive" title="b1">pp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">gg</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1 holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">pp</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c25">b2</font>;<br/>
<a NAME="E33:21_3_1"/><span class="lab"><font color="Green" title="E37">A70</font></span>: 
for <font color="Olive" title="b1">x</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="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c36">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c36">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c37">pp9</font> = <font color="Maroon" title="c35">gg</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c36">x</font> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c38">y</font> = <font color="Maroon" title="c37">pp9</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c25">b2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c36">x</font>,<font color="Maroon" title="c38">y</font>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c39">pp</font> be   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">o2</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"> ( <font color="Maroon" title="c36">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">r</font> &amp; <font color="Maroon" title="c39">pp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">gg</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c36">x</font> implies <font color="Maroon" title="c38">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">pp</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c25">b2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:21_3_1_6"/>
<font color="Maroon" title="c36">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">r</font>
 <span class="kw">and </span><br/><a NAME="E3:21_3_1_6"/><span class="lab"><font color="Green" title="E38">A71</font></span>: 
<font color="Maroon" title="c39">pp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">gg</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c36">x</font>
 ; <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="c38">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">pp</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c25">b2</font></span><br/>

<span class="kw">thus </span><a NAME="E4:21_3_1_6"/>
<font color="Maroon" title="c38">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">pp</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c25">b2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_3_1_6"><span class="lab"><font color="Green" title="E38">A71</font></span></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>
<span class="kw">consider </span><font color="Maroon" title="c36">ff</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E35:21_3_1"/><span class="lab"><font color="Green" title="E38">A72</font></span>: 
for <font color="Olive" title="b1">j</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>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">j</font>,<font color="Maroon" title="c36">ff</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">j</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E33:21_3_1"><span class="lab"><font color="Green" title="E37">A70</font></span></a>);<br/></span>
<span class="kw">defpred </span><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">a19</font>, <font color="Olive" title="b2">b19</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">o1</font> ex <font color="Olive" title="b3">Fk</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b3">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c26">b1</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Olive" title="b1">a19</font>,<font color="Olive" title="b2">b19</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> &amp; ( for <font color="Olive" title="b4">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b3">Fk</font> holds <br/> ex <font color="Olive" title="b5">a199</font>, <font color="Olive" title="b6">b199</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Olive" title="b5">a199</font>,<font color="Olive" title="b6">b199</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp; <font color="Olive" title="b3">Fk</font> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">a19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b5">a199</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">b19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b6">b199</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> ) ) );<br/>
<a NAME="E36:21_3_1"/><span class="lab"><font color="Green" title="E39">A73</font></span>: 
for <font color="Olive" title="b1">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>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">r</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default">2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c37">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c37">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">r</font></span>)</span> implies  ex <font color="Olive" title="b1">x</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default">2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c37">k</font>,<font color="Olive" title="b1">x</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_3_1_7"/>
<font color="Maroon" title="c37">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">r</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default">2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c37">k</font>,<font color="Olive" title="b1">x</font>]</span><br/>

<a NAME="E2:21_3_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c37">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c26">b1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_3_1"><span class="lab"><font color="Green" title="E22">A21</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c38">a19</font>, <font color="Maroon" title="c39">b19</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">o1</font><span class="kw"> such that </span><br/><a NAME="E4:21_3_1_7"/><span class="lab"><font color="Green" title="E40">A74</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c26">b1</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c37">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c38">a19</font>,<font color="Maroon" title="c39">b19</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E5:21_3_1_7"/>
<font color="Maroon" title="c26">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c38">a19</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c39">b19</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#T68" title="PRE_POLY:th.68">PRE_POLY:68</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c40">a19</font> = <font color="Maroon" title="c38">a19</font>, <font color="Maroon" title="c41">b19</font> = <font color="Maroon" title="c39">b19</font> as    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">o1</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/>
<span class="kw">defpred </span><font color="Maroon">S<sub>5</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">a199</font>, <font color="Olive" title="b2">b199</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Olive" title="b1">a199</font>,<font color="Olive" title="b2">b199</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c40">a19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b1">a199</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c41">b19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">b199</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> );<br/>
<a NAME="E7:21_3_1_7"/><span class="lab"><font color="Green" title="E41">A75</font></span>: 
for <font color="Olive" title="b1">k</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>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span></span>)</span> holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span> st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_1_7_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c42">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c42">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span></span>)</span> implies  ex <font color="Olive" title="b1">x</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span> st <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c42">k</font>,<font color="Olive" title="b1">x</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_3_1_7_1"/>
<font color="Maroon" title="c42">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span> st <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c42">k</font>,<font color="Olive" title="b1">x</font>]</span><br/>

<a NAME="E2:21_3_1_7_1"/><span class="kw">then </span>
<font color="Maroon" title="c42">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c43">a199</font>, <font color="Maroon" title="c44">b199</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">o2</font><span class="kw"> such that </span><br/><a NAME="E4:21_3_1_7_1"/><span class="lab"><font color="Green" title="E42">A76</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c42">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c43">a199</font>,<font color="Maroon" title="c44">b199</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E5:21_3_1_7_1"/>
<font color="Maroon" title="c25">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c43">a199</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c44">b199</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#T68" title="PRE_POLY:th.68">PRE_POLY:68</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c45">a199</font> = <font color="Maroon" title="c43">a199</font>, <font color="Maroon" title="c46">b199</font> = <font color="Maroon" title="c44">b199</font> as    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</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/>
<span class="kw">reconsider </span><font color="Maroon" title="c47">x</font> = <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c40">a19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c45">a199</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c41">b19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c46">b199</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c47">x</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c42">k</font>,<font color="Maroon" title="c47">x</font>]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c45">a199</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">b199</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c42">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c45">a199</font>,<font color="Olive" title="b1">b199</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp; <font color="Maroon" title="c47">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c40">a19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c45">a199</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c41">b19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b1">b199</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c46">b199</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"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c42">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c45">a199</font>,<font color="Maroon" title="c46">b199</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp; <font color="Maroon" title="c47">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c40">a19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c45">a199</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c41">b19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c46">b199</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> )</span><br/>

<span class="kw">thus </span><a NAME="E8:21_3_1_7_1"/>
( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c42">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c45">a199</font>,<font color="Maroon" title="c46">b199</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp; <font color="Maroon" title="c47">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c40">a19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c45">a199</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c41">b19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c46">b199</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_3_1_7_1"><span class="lab"><font color="Green" title="E42">A76</font></span></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>
<span class="kw">consider </span><font color="Maroon" title="c42">Fk</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E9:21_3_1_7"/><span class="lab"><font color="Green" title="E42">A77</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c42">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:21_3_1_7"/><span class="lab"><font color="Green" title="E43">A78</font></span>: 
for <font color="Olive" title="b1">k</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>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c42">Fk</font> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">k</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="recdef_1.html#S17" title="RECDEF_1:sch.17">RECDEF_1:sch 17</a>(<a class="txt" href="#E7:21_3_1_7"><span class="lab"><font color="Green" title="E41">A75</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c43">x</font> = <font color="Maroon" title="c42">Fk</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default">2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c43">x</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c37">k</font>,<font color="Maroon" title="c43">x</font>]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c40">a19</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">b19</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">o1</font> ex <font color="Olive" title="b2">Fk</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b2">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c43">x</font> &amp; <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c26">b1</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c37">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c40">a19</font>,<font color="Olive" title="b1">b19</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> &amp; ( for <font color="Olive" title="b3">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b2">Fk</font> holds <br/> ex <font color="Olive" title="b4">a199</font>, <font color="Olive" title="b5">b199</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Olive" title="b4">a199</font>,<font color="Olive" title="b5">b199</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp; <font color="Olive" title="b2">Fk</font> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c40">a19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b4">a199</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">b19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b5">b199</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> ) ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c41">b19</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">Fk</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b1">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c43">x</font> &amp; <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c26">b1</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c37">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c40">a19</font>,<font color="Maroon" title="c41">b19</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> &amp; ( for <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">Fk</font> holds <br/> ex <font color="Olive" title="b3">a199</font>, <font color="Olive" title="b4">b199</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Olive" title="b3">a199</font>,<font color="Olive" title="b4">b199</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp; <font color="Olive" title="b1">Fk</font> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c40">a19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b3">a199</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c41">b19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b4">b199</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> ) ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c42">Fk</font>
; <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="c42">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c43">x</font> &amp; <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c26">b1</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c37">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c40">a19</font>,<font color="Maroon" title="c41">b19</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> &amp; ( for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c42">Fk</font> holds <br/> ex <font color="Olive" title="b2">a199</font>, <font color="Olive" title="b3">b199</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Olive" title="b2">a199</font>,<font color="Olive" title="b3">b199</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp; <font color="Maroon" title="c42">Fk</font> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c40">a19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">a199</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c41">b19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b3">b199</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E12:21_3_1_7"/>
<font color="Maroon" title="c42">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c43">x</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"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c26">b1</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c37">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c40">a19</font>,<font color="Maroon" title="c41">b19</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> &amp; ( for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c42">Fk</font> holds <br/> ex <font color="Olive" title="b2">a199</font>, <font color="Olive" title="b3">b199</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Olive" title="b2">a199</font>,<font color="Olive" title="b3">b199</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp; <font color="Maroon" title="c42">Fk</font> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c40">a19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">a199</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c41">b19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b3">b199</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E13:21_3_1_7"/>
<span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c26">b1</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c37">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c40">a19</font>,<font color="Maroon" title="c41">b19</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_3_1_7"><span class="lab"><font color="Green" title="E40">A74</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> &amp; ( for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c42">Fk</font> holds <br/> ex <font color="Olive" title="b2">a199</font>, <font color="Olive" title="b3">b199</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Olive" title="b2">a199</font>,<font color="Olive" title="b3">b199</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp; <font color="Maroon" title="c42">Fk</font> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c40">a19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">a199</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c41">b19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b3">b199</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E14:21_3_1_7"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:21_3_1_7"><span class="lab"><font color="Green" title="E42">A77</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>; <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">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c42">Fk</font> holds <br/> ex <font color="Olive" title="b2">a199</font>, <font color="Olive" title="b3">b199</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Olive" title="b2">a199</font>,<font color="Olive" title="b3">b199</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp; <font color="Maroon" title="c42">Fk</font> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c40">a19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">a199</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c41">b19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b3">b199</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c44">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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="c44">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c42">Fk</font> implies  ex <font color="Olive" title="b1">a199</font>, <font color="Olive" title="b2">b199</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c44">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Olive" title="b1">a199</font>,<font color="Olive" title="b2">b199</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp; <font color="Maroon" title="c42">Fk</font> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c44">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c40">a19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b1">a199</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c41">b19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">b199</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E15:21_3_1_7"/>
<font color="Maroon" title="c44">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c42">Fk</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a199</font>, <font color="Olive" title="b2">b199</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c44">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Olive" title="b1">a199</font>,<font color="Olive" title="b2">b199</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp; <font color="Maroon" title="c42">Fk</font> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c44">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c40">a19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b1">a199</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c41">b19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">b199</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> )</span><br/>

<span class="kw">hence </span><a NAME="E16:21_3_1_7"/>
 ex <font color="Olive" title="b1">a199</font>, <font color="Olive" title="b2">b199</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c44">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Olive" title="b1">a199</font>,<font color="Olive" title="b2">b199</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp; <font color="Maroon" title="c42">Fk</font> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c44">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c40">a19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b1">a199</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c41">b19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">b199</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:21_3_1_7"><span class="lab"><font color="Green" title="E42">A77</font></span></a>, <a class="txt" href="#E10:21_3_1_7"><span class="lab"><font color="Green" title="E43">A78</font></span></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>
<span class="kw">consider </span><font color="Maroon" title="c37">G</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <span class="p1">(<span class="default">2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> <span class="kw"> such that </span><br/><a NAME="E38:21_3_1"/><span class="lab"><font color="Green" title="E40">A79</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c37">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">r</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E39:21_3_1"/><span class="lab"><font color="Green" title="E41">A80</font></span>: 
for <font color="Olive" title="b1">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>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">r</font></span>)</span> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c37">G</font> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">i</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="recdef_1.html#S17" title="RECDEF_1:sch.17">RECDEF_1:sch 17</a>(<a class="txt" href="#E36:21_3_1"><span class="lab"><font color="Green" title="E39">A73</font></span></a>);<br/></span>
<a NAME="E40:21_3_1"/><span class="lab"><font color="Green" title="E42">A81</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">r</font></span>)</span> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c37">G</font> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">i</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:21_3_1"><span class="lab"><font color="Green" title="E41">A80</font></span></a></span>;<br/>
<a NAME="E41:21_3_1"/><span class="lab"><font color="Green" title="E43">A82</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <font color="Maroon" title="c31">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c31">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a></span>;<br/>
<a NAME="E42:21_3_1"/><span class="lab"><font color="Green" title="E44">A83</font></span>: 
for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <font color="Maroon" title="c31">F</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <font color="Maroon" title="c31">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <font color="Maroon" title="c37">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_1_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c38">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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="c38">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <font color="Maroon" title="c31">F</font></span>)</span> implies <span class="p1">(<span class="default"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <font color="Maroon" title="c31">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c38">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <font color="Maroon" title="c37">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c38">j</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_3_1_8"/><span class="lab"><font color="Green" title="E45">A84</font></span>: 
<font color="Maroon" title="c38">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <font color="Maroon" title="c31">F</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"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <font color="Maroon" title="c31">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c38">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <font color="Maroon" title="c37">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c38">j</font></span><br/>

<a NAME="E2:21_3_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A85</font></span>: 
<font color="Maroon" title="c38">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c31">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a></span>;<br/>
<a NAME="E3:21_3_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A86</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <font color="Maroon" title="c31">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c38">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c38">j</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a></span>;<br/>
<a NAME="E4:21_3_1_8"/><span class="lab"><font color="Green" title="E48">A87</font></span>: 
(  ex <font color="Olive" title="b1">a19</font>, <font color="Olive" title="b2">b19</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">o1</font> ex <font color="Olive" title="b3">Fk</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font> ex <font color="Olive" title="b4">pa19</font>, <font color="Olive" title="b5">qb19</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font> st <br/>( <font color="Olive" title="b4">pa19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">p</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b1">a19</font> &amp; <font color="Olive" title="b5">qb19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b2">b19</font> &amp; <font color="Olive" title="b3">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">F</font> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c38">j</font> &amp; <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c26">b1</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c38">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b1">a19</font>,<font color="Olive" title="b2">b19</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> &amp; ( for <font color="Olive" title="b6">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b3">Fk</font> holds <br/> ex <font color="Olive" title="b7">a199</font>, <font color="Olive" title="b8">b199</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b7">a199</font>,<font color="Olive" title="b8">b199</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Olive" title="b3">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">pa19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b7">a199</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">qb19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b8">b199</font></span>)</span> ) ) ) &amp;  ex <font color="Olive" title="b1">a29</font>, <font color="Olive" title="b2">b29</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">o1</font> ex <font color="Olive" title="b3">Gk</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b3">Gk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">G</font> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c38">j</font> &amp; <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c26">b1</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c38">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Olive" title="b1">a29</font>,<font color="Olive" title="b2">b29</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">Gk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> &amp; ( for <font color="Olive" title="b4">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b3">Gk</font> holds <br/> ex <font color="Olive" title="b5">a299</font>, <font color="Olive" title="b6">b299</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Olive" title="b5">a299</font>,<font color="Olive" title="b6">b299</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp; <font color="Olive" title="b3">Gk</font> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">a29</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b5">a299</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">b29</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b6">b299</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> ) ) ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:21_3_1"><span class="lab"><font color="Green" title="E28">A39</font></span></a>, <a class="txt" href="#E21:21_3_1"><span class="lab"><font color="Green" title="E29">A40</font></span></a>, <a class="txt" href="#E39:21_3_1"><span class="lab"><font color="Green" title="E41">A80</font></span></a>, <a class="txt" href="#E2:21_3_1_8"><span class="lab"><font color="Green" title="E46">A85</font></span></a></span>;<br/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c38">j</font></span>)</span> = 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">F</font> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c38">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_3_1_8"><span class="lab"><font color="Green" title="E46">A85</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c38">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:21_3_1"><span class="lab"><font color="Green" title="E28">A39</font></span></a>, <a class="txt" href="#E38:21_3_1"><span class="lab"><font color="Green" title="E40">A79</font></span></a>, <a class="txt" href="#E41:21_3_1"><span class="lab"><font color="Green" title="E43">A82</font></span></a>, <a class="txt" href="#E1:21_3_1_8"><span class="lab"><font color="Green" title="E45">A84</font></span></a>, <a class="txt" href="#E4:21_3_1_8"><span class="lab"><font color="Green" title="E48">A87</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:21_3_1_8"/>
<span class="p1">(<span class="default"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <font color="Maroon" title="c31">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c38">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <font color="Maroon" title="c37">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c38">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:21_3_1"><span class="lab"><font color="Green" title="E28">A39</font></span></a>, <a class="txt" href="#E38:21_3_1"><span class="lab"><font color="Green" title="E40">A79</font></span></a>, <a class="txt" href="#E41:21_3_1"><span class="lab"><font color="Green" title="E43">A82</font></span></a>, <a class="txt" href="#E1:21_3_1_8"><span class="lab"><font color="Green" title="E45">A84</font></span></a>, <a class="txt" href="#E3:21_3_1_8"><span class="lab"><font color="Green" title="E47">A86</font></span></a>, <a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</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>
<span class="kw">consider </span><font color="Maroon" title="c38">c1</font> being    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c39">c2</font> being    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font>, <font color="Maroon" title="c40">Q1</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E44:21_3_1"/><span class="lab"><font color="Green" title="E45">A88</font></span>: 
<font color="Maroon" title="c40">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c15">q</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c38">c1</font>
 <span class="kw">and </span><br/><a NAME="E45:21_3_1"/><span class="lab"><font color="Green" title="E46">A89</font></span>: 
<font color="Maroon" title="c23">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c38">c1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c39">c2</font>
 <span class="kw">and </span><br/><a NAME="E46:21_3_1"/><span class="lab"><font color="Green" title="E47">A90</font></span>: 
<span class="p1">(<span class="default"><a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c15">q</font></span>)</span></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c23">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c40">Q1</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c39">c2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom6.html#D2" target="_self" title="POLYNOM6:def.2">Def2</a></span>;<br/>
<a NAME="E47:21_3_1"/><span class="lab"><font color="Green" title="E48">A91</font></span>: 
<font color="Maroon" title="c38">c1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c26">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_3_1"><span class="lab"><font color="Green" title="E20">A19</font></span></a>, <a class="txt" href="#E45:21_3_1"><span class="lab"><font color="Green" title="E46">A89</font></span></a>, <a class="ref" href="polynom6.html#T7" target="_self" title="POLYNOM6:th.7">Th7</a></span>;<br/>
<a NAME="E48:21_3_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c37">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c38">c1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_3_1"><span class="lab"><font color="Green" title="E22">A21</font></span></a>, <a class="txt" href="#E38:21_3_1"><span class="lab"><font color="Green" title="E40">A79</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E49:21_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A92</font></span>: 
 <a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c22">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c37">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_3_1"><span class="lab"><font color="Green" title="E20">A19</font></span></a>, <a class="txt" href="#E38:21_3_1"><span class="lab"><font color="Green" title="E40">A79</font></span></a>, <a class="txt" href="#E40:21_3_1"><span class="lab"><font color="Green" title="E42">A81</font></span></a>, <a class="txt" href="#E47:21_3_1"><span class="lab"><font color="Green" title="E48">A91</font></span></a>, <a class="ref" href="polynom6.html#T14" target="_self" title="POLYNOM6:th.14">Th14</a></span>;<br/>
<a NAME="E50:21_3_1"/><span class="lab"><font color="Green" title="E50">A93</font></span>: 
for <font color="Olive" title="b1">j</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="b2">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font>  st <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c33">rFFb2</font> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">rFFb2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> holds <br/><font color="Maroon" title="c36">ff</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">ff</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_1_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c41">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font>  st <font color="Maroon" title="c41">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c33">rFFb2</font> &amp; <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">rFFb2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> holds <br/><font color="Maroon" title="c36">ff</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">ff</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c41">j</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b1">v</font></span><br/><span class="kw">let </span><font color="Maroon" title="c42">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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"> ( <font color="Maroon" title="c41">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c33">rFFb2</font> &amp; <font color="Maroon" title="c42">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">rFFb2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> implies <font color="Maroon" title="c36">ff</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">ff</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c41">j</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c42">v</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:21_3_1_9"/><span class="lab"><font color="Green" title="E51">A94</font></span>: 
<font color="Maroon" title="c41">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c33">rFFb2</font>
 <span class="kw">and </span><br/><a NAME="E2:21_3_1_9"/><span class="lab"><font color="Green" title="E52">A95</font></span>: 
<font color="Maroon" title="c42">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">rFFb2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <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="c36">ff</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">ff</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c41">j</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c42">v</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c43">w</font> = <font color="Maroon" title="c27">r</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>, <font color="Maroon" title="c44">pp</font> = <font color="Maroon" title="c35">gg</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c41">j</font>, <font color="Maroon" title="c45">pp9</font> = <font color="Maroon" title="c35">gg</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c46">w1</font> = <font color="Maroon" title="c43">w</font>, <font color="Maroon" title="c47">pp1</font> = <font color="Maroon" title="c44">pp</font>, <font color="Maroon" title="c48">pp19</font> = <font color="Maroon" title="c45">pp9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c49">w1</font> = <font color="Maroon" title="c46">w1</font>, <font color="Maroon" title="c50">pp1</font> = <font color="Maroon" title="c47">pp1</font>, <font color="Maroon" title="c51">pp19</font> = <font color="Maroon" title="c48">pp19</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span> ;<br/>
<a NAME="E6:21_3_1_9"/><span class="lab"><font color="Green" title="E53">A96</font></span>: 
<font color="Maroon" title="c41">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21_3_1"><span class="lab"><font color="Green" title="E24">A24</font></span></a>, <a class="txt" href="#E1:21_3_1_9"><span class="lab"><font color="Green" title="E51">A94</font></span></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>;<br/>
<a NAME="E7:21_3_1_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A97</font></span>: 
<font color="Maroon" title="c41">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E8:21_3_1_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A98</font></span>: 
<font color="Maroon" title="c43">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E9:21_3_1_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A99</font></span>: 
<font color="Maroon" title="c51">pp19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c50">pp1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c49">w1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:21_3_1"><span class="lab"><font color="Green" title="E36">A69</font></span></a>, <a class="txt" href="#E6:21_3_1_9"><span class="lab"><font color="Green" title="E53">A96</font></span></a></span>;<br/>
<a NAME="E10:21_3_1_9"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c41">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E11:21_3_1_9"/><span class="kw">then </span>
<font color="Maroon" title="c41">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c27">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_3_1_9"><span class="lab"><font color="Green" title="E54">A97</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E12:21_3_1_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A100</font></span>: 
<font color="Maroon" title="c43">w</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c25">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:21_3_1"><span class="lab"><font color="Green" title="E25">A25</font></span></a>, <a class="txt" href="#E2:21_3_1_9"><span class="lab"><font color="Green" title="E52">A95</font></span></a>, <a class="txt" href="#E8:21_3_1_9"><span class="lab"><font color="Green" title="E55">A98</font></span></a></span>;<br/>
<a NAME="E13:21_3_1_9"/>
<font color="Maroon" title="c41">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21_3_1_9"><span class="lab"><font color="Green" title="E53">A96</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c36">ff</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<font color="Maroon" title="c45">pp9</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c25">b2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:21_3_1"><span class="lab"><font color="Green" title="E38">A72</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c44">pp</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">+</a> <font color="Maroon" title="c43">w</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c25">b2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:21_3_1_9"><span class="lab"><font color="Green" title="E56">A99</font></span></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c44">pp</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c43">w</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c25">b2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D21" title="POLYNOM1:def.21">POLYNOM1:def 21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c36">ff</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c41">j</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c42">v</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:21_3_1"><span class="lab"><font color="Green" title="E38">A72</font></span></a>, <a class="txt" href="#E6:21_3_1_9"><span class="lab"><font color="Green" title="E53">A96</font></span></a>, <a class="txt" href="#E12:21_3_1_9"><span class="lab"><font color="Green" title="E57">A100</font></span></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>
<a NAME="E51:21_3_1"/>
<font color="Maroon" title="c35">gg</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:21_3_1"><span class="lab"><font color="Green" title="E35">A68</font></span></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E51">A101</font></span>: <font color="Maroon" title="c36">ff</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<span class="p1">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c25">b2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:21_3_1"><span class="lab"><font color="Green" title="E38">A72</font></span></a>, <a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></span>
<br/>.= 
 <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="polynom1.html#T81" title="POLYNOM1:th.81">POLYNOM1:81</a></span>
;<br/>
<a NAME="E53:21_3_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c33">rFFb2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21_3_1"><span class="lab"><font color="Green" title="E24">A24</font></span></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>;<br/>
<a NAME="E54:21_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c34">Sr</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c25">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">ff</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c33">rFFb2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:21_3_1"><span class="lab"><font color="Green" title="E34">A67</font></span></a>, <a class="txt" href="#E35:21_3_1"><span class="lab"><font color="Green" title="E38">A72</font></span></a></span>;<br/>
<a NAME="E55:21_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A102</font></span>: 
<font color="Maroon" title="c34">Sr</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c25">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c33">rFFb2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:21_3_1"><span class="lab"><font color="Green" title="E51">A101</font></span></a>, <a class="txt" href="#E50:21_3_1"><span class="lab"><font color="Green" title="E50">A93</font></span></a>, <a class="ref" href="rlvect_1.html#D15" title="RLVECT_1:def.15">RLVECT_1:def 15</a></span>;<br/>
<a NAME="E56:21_3_1"/>
( <font color="Maroon" title="c26">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c38">c1</font> &amp; <font color="Maroon" title="c25">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c39">c2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_3_1"><span class="lab"><font color="Green" title="E20">A19</font></span></a>, <a class="txt" href="#E45:21_3_1"><span class="lab"><font color="Green" title="E46">A89</font></span></a>, <a class="ref" href="polynom6.html#T7" target="_self" title="POLYNOM6:th.7">Th7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E57:21_3_1"/>
<font color="Maroon" title="c20">fpq9</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c22">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c32">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_3_1"><span class="lab"><font color="Green" title="E21">A20</font></span></a>, <a class="txt" href="#E44:21_3_1"><span class="lab"><font color="Green" title="E45">A88</font></span></a>, <a class="txt" href="#E46:21_3_1"><span class="lab"><font color="Green" title="E47">A90</font></span></a>, <a class="txt" href="#E27:21_3_1"><span class="lab"><font color="Green" title="E33">A66</font></span></a>, <a class="txt" href="#E55:21_3_1"><span class="lab"><font color="Green" title="E52">A102</font></span></a>, <a class="txt" href="#E24:21_3_1"><span class="lab"><font color="Green" title="E31">A42</font></span></a>, <a class="ref" href="polynom1.html#T34" title="POLYNOM1:th.34">POLYNOM1:34</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c22">c</font></span>)</span> &amp; ( for <font color="Olive" title="b1">k</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>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c32">s</font> holds <br/> ex <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">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c22">c</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b2">b1</font>,<font color="Olive" title="b3">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c32">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">fp</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b2">b1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">fq</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b3">b2</font></span>)</span> ) ) )</span><br/>

<a NAME="E58:21_3_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <font color="Maroon" title="c37">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c37">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a></span>;<br/>
<a NAME="E59:21_3_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <font color="Maroon" title="c31">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <font color="Maroon" title="c37">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:21_3_1"><span class="lab"><font color="Green" title="E28">A39</font></span></a>, <a class="txt" href="#E38:21_3_1"><span class="lab"><font color="Green" title="E40">A79</font></span></a>, <a class="txt" href="#E41:21_3_1"><span class="lab"><font color="Green" title="E43">A82</font></span></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>;<br/>
<a NAME="E60:21_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A103</font></span>: 
 <a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <font color="Maroon" title="c31">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <font color="Maroon" title="c37">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:21_3_1"><span class="lab"><font color="Green" title="E44">A83</font></span></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E61:21_3_1"/><span class="lab"><font color="Green" title="E54">A104</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c22">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:21_3_1"><span class="lab"><font color="Green" title="E49">A92</font></span></a>, <a class="ref" href="pre_poly.html#T28" title="PRE_POLY:th.28">PRE_POLY:28</a></span>; <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">k</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>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c32">s</font> holds <br/> ex <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">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c22">c</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b2">b1</font>,<font color="Olive" title="b3">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c32">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">fp</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b2">b1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">fq</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b3">b2</font></span>)</span> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c41">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c41">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c32">s</font> implies  ex <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">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c22">c</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c41">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b1">b1</font>,<font color="Olive" title="b2">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c32">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c41">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">fp</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b1">b1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">fq</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b2">b2</font></span>)</span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E62:21_3_1"/><span class="lab"><font color="Green" title="E55">A105</font></span>: 
<font color="Maroon" title="c41">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c32">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <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">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c22">c</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c41">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b1">b1</font>,<font color="Olive" title="b2">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c32">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c41">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">fp</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b1">b1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">fq</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b2">b2</font></span>)</span> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c42">i</font>, <font color="Maroon" title="c43">j</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> <span class="kw"> such that </span><br/><a NAME="E64:21_3_1"/><span class="lab"><font color="Green" title="E56">A106</font></span>: 
<font color="Maroon" title="c42">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c31">F</font>
 <span class="kw">and </span><br/><a NAME="E65:21_3_1"/><span class="lab"><font color="Green" title="E57">A107</font></span>: 
<font color="Maroon" title="c43">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">i</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E66:21_3_1"/><span class="lab"><font color="Green" title="E58">A108</font></span>: 
<font color="Maroon" title="c41">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c31">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c42">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">j</font>
 <span class="kw">and </span><br/><a NAME="E67:21_3_1"/><span class="lab"><font color="Green" title="E59">A109</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c31">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c31">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#T29" title="PRE_POLY:th.29">PRE_POLY:29</a></span>;<br/>
<a NAME="E68:21_3_1"/><span class="lab"><font color="Green" title="E60">A110</font></span>: 
<font color="Maroon" title="c41">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c22">c</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E61:21_3_1"><span class="lab"><font color="Green" title="E54">A104</font></span></a>, <a class="txt" href="#E62:21_3_1"><span class="lab"><font color="Green" title="E55">A105</font></span></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c44">i9</font>, <font color="Maroon" title="c45">j9</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> <span class="kw"> such that </span><br/><a NAME="E70:21_3_1"/><span class="lab"><font color="Green" title="E61">A111</font></span>: 
<font color="Maroon" title="c44">i9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c37">G</font>
 <span class="kw">and </span><br/><a NAME="E71:21_3_1"/><span class="lab"><font color="Green" title="E62">A112</font></span>: 
<font color="Maroon" title="c45">j9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i9</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E72:21_3_1"/><span class="lab"><font color="Green" title="E63">A113</font></span>: 
<font color="Maroon" title="c41">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c37">G</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c44">i9</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c45">j9</font>
 <span class="kw">and </span><br/><a NAME="E73:21_3_1"/><span class="lab"><font color="Green" title="E64">A114</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c37">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i9</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">j9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c22">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:21_3_1"><span class="lab"><font color="Green" title="E49">A92</font></span></a>, <a class="ref" href="pre_poly.html#T29" title="PRE_POLY:th.29">PRE_POLY:29</a></span>;<br/>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <font color="Maroon" title="c31">F</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c42">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">j</font> = 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c31">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c42">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">j</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#T16" title="POLYNOM3:th.16">POLYNOM3:16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <font color="Maroon" title="c37">G</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c44">i9</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c45">j9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:21_3_1"><span class="lab"><font color="Green" title="E58">A108</font></span></a>, <a class="txt" href="#E72:21_3_1"><span class="lab"><font color="Green" title="E63">A113</font></span></a>, <a class="ref" href="polynom3.html#T16" title="POLYNOM3:th.16">POLYNOM3:16</a></span>
;<br/>
<a NAME="E75:21_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A115</font></span>: 
( <font color="Maroon" title="c42">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">i9</font> &amp; <font color="Maroon" title="c43">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c45">j9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:21_3_1"><span class="lab"><font color="Green" title="E53">A103</font></span></a>, <a class="txt" href="#E64:21_3_1"><span class="lab"><font color="Green" title="E56">A106</font></span></a>, <a class="txt" href="#E65:21_3_1"><span class="lab"><font color="Green" title="E57">A107</font></span></a>, <a class="txt" href="#E70:21_3_1"><span class="lab"><font color="Green" title="E61">A111</font></span></a>, <a class="txt" href="#E71:21_3_1"><span class="lab"><font color="Green" title="E62">A112</font></span></a>, <a class="ref" href="polynom3.html#T22" title="POLYNOM3:th.22">POLYNOM3:22</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c46">c1</font>, <font color="Maroon" title="c47">c2</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font><span class="kw"> such that </span><br/><a NAME="E77:21_3_1"/><span class="lab"><font color="Green" title="E66">A116</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c22">c</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c41">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c46">c1</font>,<font color="Maroon" title="c47">c2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E78:21_3_1"/>
<font color="Maroon" title="c22">c</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c46">c1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c47">c2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E68:21_3_1"><span class="lab"><font color="Green" title="E60">A110</font></span></a>, <a class="ref" href="pre_poly.html#T68" title="PRE_POLY:th.68">PRE_POLY:68</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c48">cc1</font> = <font color="Maroon" title="c46">c1</font>, <font color="Maroon" title="c49">cc2</font> = <font color="Maroon" title="c47">c2</font> as    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span> <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/>
<span class="kw">consider </span><font color="Maroon" title="c50">cb1</font> being    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c51">cb2</font> being    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font>, <font color="Maroon" title="c52">Q1</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E81:21_3_1"/><span class="lab"><font color="Green" title="E67">A117</font></span>: 
<font color="Maroon" title="c52">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">p</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c50">cb1</font>
 <span class="kw">and </span><br/><a NAME="E82:21_3_1"/><span class="lab"><font color="Green" title="E68">A118</font></span>: 
<font color="Maroon" title="c48">cc1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c50">cb1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c51">cb2</font>
 <span class="kw">and </span><br/><a NAME="E83:21_3_1"/><span class="lab"><font color="Green" title="E69">A119</font></span>: 
<span class="p1">(<span class="default"><a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c14">p</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c48">cc1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c52">Q1</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c51">cb2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom6.html#D2" target="_self" title="POLYNOM6:def.2">Def2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c53">a19</font>, <font color="Maroon" title="c54">b19</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c55">Fk</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font>, <font color="Maroon" title="c56">pa19</font>, <font color="Maroon" title="c57">qb19</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E85:21_3_1"/><span class="lab"><font color="Green" title="E70">A120</font></span>: 
<font color="Maroon" title="c56">pa19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">p</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c53">a19</font>
 <span class="kw">and </span><br/><a NAME="E86:21_3_1"/><span class="lab"><font color="Green" title="E71">A121</font></span>: 
<font color="Maroon" title="c57">qb19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c54">b19</font>
 <span class="kw">and </span><br/><a NAME="E87:21_3_1"/><span class="lab"><font color="Green" title="E72">A122</font></span>: 
<font color="Maroon" title="c55">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">F</font> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c42">i</font>
 <span class="kw">and </span><br/><a NAME="E88:21_3_1"/><span class="lab"><font color="Green" title="E73">A123</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c26">b1</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c42">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c53">a19</font>,<font color="Maroon" title="c54">b19</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E89:21_3_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c55">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E90:21_3_1"/><span class="lab"><font color="Green" title="E74">A124</font></span>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c55">Fk</font> holds <br/> ex <font color="Olive" title="b2">a199</font>, <font color="Olive" title="b3">b199</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b2">a199</font>,<font color="Olive" title="b3">b199</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c55">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c56">pa19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b2">a199</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c57">qb19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b3">b199</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:21_3_1"><span class="lab"><font color="Green" title="E28">A39</font></span></a>, <a class="txt" href="#E21:21_3_1"><span class="lab"><font color="Green" title="E29">A40</font></span></a>, <a class="txt" href="#E64:21_3_1"><span class="lab"><font color="Green" title="E56">A106</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c58">ga19</font>, <font color="Maroon" title="c59">gb19</font> being    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c60">Gk</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E92:21_3_1"/><span class="lab"><font color="Green" title="E75">A125</font></span>: 
<font color="Maroon" title="c60">Gk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">G</font> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c42">i</font>
 <span class="kw">and </span><br/><a NAME="E93:21_3_1"/><span class="lab"><font color="Green" title="E76">A126</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c26">b1</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c42">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c58">ga19</font>,<font color="Maroon" title="c59">gb19</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E94:21_3_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c60">Gk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E95:21_3_1"/><span class="lab"><font color="Green" title="E77">A127</font></span>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c60">Gk</font> holds <br/> ex <font color="Olive" title="b2">ga199</font>, <font color="Olive" title="b3">gb199</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Olive" title="b2">ga199</font>,<font color="Olive" title="b3">gb199</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp; <font color="Maroon" title="c60">Gk</font> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c58">ga19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">ga199</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c59">gb19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b3">gb199</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:21_3_1"><span class="lab"><font color="Green" title="E28">A39</font></span></a>, <a class="txt" href="#E39:21_3_1"><span class="lab"><font color="Green" title="E41">A80</font></span></a>, <a class="txt" href="#E64:21_3_1"><span class="lab"><font color="Green" title="E56">A106</font></span></a></span>;<br/>
<a NAME="E96:21_3_1"/><span class="lab"><font color="Green" title="E78">A128</font></span>: 
<font color="Maroon" title="c54">b19</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c59">gb19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E88:21_3_1"><span class="lab"><font color="Green" title="E73">A123</font></span></a>, <a class="txt" href="#E93:21_3_1"><span class="lab"><font color="Green" title="E76">A126</font></span></a>, <a class="ref" href="finseq_1.html#T98" title="FINSEQ_1:th.98">FINSEQ_1:98</a></span>;<br/>
<a NAME="E97:21_3_1"/><span class="lab"><font color="Green" title="E79">A129</font></span>: 
<font color="Maroon" title="c60">Gk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:21_3_1"><span class="lab"><font color="Green" title="E28">A39</font></span></a>, <a class="txt" href="#E38:21_3_1"><span class="lab"><font color="Green" title="E40">A79</font></span></a>, <a class="txt" href="#E64:21_3_1"><span class="lab"><font color="Green" title="E56">A106</font></span></a>, <a class="txt" href="#E92:21_3_1"><span class="lab"><font color="Green" title="E75">A125</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c61">ga199</font>, <font color="Maroon" title="c62">gb199</font> being    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font><span class="kw"> such that </span><br/><a NAME="E99:21_3_1"/><span class="lab"><font color="Green" title="E80">A130</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c43">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c61">ga199</font>,<font color="Maroon" title="c62">gb199</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E100:21_3_1"/><span class="lab"><font color="Green" title="E81">A131</font></span>: 
<font color="Maroon" title="c60">Gk</font> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c43">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c58">ga19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c61">ga199</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c59">gb19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c62">gb199</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E71:21_3_1"><span class="lab"><font color="Green" title="E62">A112</font></span></a>, <a class="txt" href="#E75:21_3_1"><span class="lab"><font color="Green" title="E65">A115</font></span></a>, <a class="txt" href="#E95:21_3_1"><span class="lab"><font color="Green" title="E77">A127</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E82">A132</font></span>: <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c46">c1</font>,<font color="Maroon" title="c47">c2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c37">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E68:21_3_1"><span class="lab"><font color="Green" title="E60">A110</font></span></a>, <a class="txt" href="#E77:21_3_1"><span class="lab"><font color="Green" title="E66">A116</font></span></a>, <a class="txt" href="#E73:21_3_1"><span class="lab"><font color="Green" title="E64">A114</font></span></a>, <a class="txt" href="#E75:21_3_1"><span class="lab"><font color="Green" title="E65">A115</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
<span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c58">ga19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c61">ga199</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c59">gb19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c62">gb199</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E71:21_3_1"><span class="lab"><font color="Green" title="E62">A112</font></span></a>, <a class="txt" href="#E75:21_3_1"><span class="lab"><font color="Green" title="E65">A115</font></span></a>, <a class="txt" href="#E97:21_3_1"><span class="lab"><font color="Green" title="E79">A129</font></span></a>, <a class="txt" href="#E100:21_3_1"><span class="lab"><font color="Green" title="E81">A131</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
;<br/>
<a NAME="E102:21_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c46">c1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c58">ga19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c61">ga199</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T98" title="FINSEQ_1:th.98">FINSEQ_1:98</a></span>;<br/>
<a NAME="E103:21_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A133</font></span>: 
( <font color="Maroon" title="c50">cb1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c58">ga19</font> &amp; <font color="Maroon" title="c51">cb2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c61">ga199</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E82:21_3_1"><span class="lab"><font color="Green" title="E68">A118</font></span></a>, <a class="ref" href="polynom6.html#T7" target="_self" title="POLYNOM6:th.7">Th7</a></span>;<br/>
<a NAME="E104:21_3_1"/><span class="lab"><font color="Green" title="E84">A134</font></span>: 
<font color="Maroon" title="c53">a19</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c58">ga19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E88:21_3_1"><span class="lab"><font color="Green" title="E73">A123</font></span></a>, <a class="txt" href="#E93:21_3_1"><span class="lab"><font color="Green" title="E76">A126</font></span></a>, <a class="ref" href="finseq_1.html#T98" title="FINSEQ_1:th.98">FINSEQ_1:98</a></span>;<br/>
<a NAME="E105:21_3_1"/>
<font color="Maroon" title="c43">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c55">Fk</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:21_3_1"><span class="lab"><font color="Green" title="E56">A106</font></span></a>, <a class="txt" href="#E65:21_3_1"><span class="lab"><font color="Green" title="E57">A107</font></span></a>, <a class="txt" href="#E87:21_3_1"><span class="lab"><font color="Green" title="E72">A122</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c63">a199</font>, <font color="Maroon" title="c64">b199</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">o2</font><span class="kw"> such that </span><br/><a NAME="E107:21_3_1"/><span class="lab"><font color="Green" title="E85">A135</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c25">b2</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c43">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c63">a199</font>,<font color="Maroon" title="c64">b199</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E108:21_3_1"/><span class="lab"><font color="Green" title="E86">A136</font></span>: 
<font color="Maroon" title="c55">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c43">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c56">pa19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c63">a199</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c57">qb19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c64">b199</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E90:21_3_1"><span class="lab"><font color="Green" title="E74">A124</font></span></a></span>;<br/>
<a NAME="E109:21_3_1"/>
<font color="Maroon" title="c63">a199</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c61">ga199</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E99:21_3_1"><span class="lab"><font color="Green" title="E80">A130</font></span></a>, <a class="txt" href="#E107:21_3_1"><span class="lab"><font color="Green" title="E85">A135</font></span></a>, <a class="ref" href="finseq_1.html#T98" title="FINSEQ_1:th.98">FINSEQ_1:98</a></span>;<br/>
<a NAME="E110:21_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A137</font></span>: 
<font color="Maroon" title="c56">pa19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c63">a199</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">fp</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c46">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E85:21_3_1"><span class="lab"><font color="Green" title="E70">A120</font></span></a>, <a class="txt" href="#E81:21_3_1"><span class="lab"><font color="Green" title="E67">A117</font></span></a>, <a class="txt" href="#E83:21_3_1"><span class="lab"><font color="Green" title="E69">A119</font></span></a>, <a class="txt" href="#E103:21_3_1"><span class="lab"><font color="Green" title="E83">A133</font></span></a>, <a class="txt" href="#E104:21_3_1"><span class="lab"><font color="Green" title="E84">A134</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c46">c1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">b2</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c22">c</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c41">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c46">c1</font>,<font color="Olive" title="b1">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c32">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c41">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">fp</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c46">c1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">fq</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b1">b2</font></span>)</span> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c47">c2</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"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c22">c</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c41">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c46">c1</font>,<font color="Maroon" title="c47">c2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c32">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c41">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">fp</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c46">c1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">fq</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c47">c2</font></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E111:21_3_1"/>
<span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c22">c</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c41">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c46">c1</font>,<font color="Maroon" title="c47">c2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E77:21_3_1"><span class="lab"><font color="Green" title="E66">A116</font></span></a></span>; <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="c32">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c41">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">fp</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c46">c1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">fq</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c47">c2</font></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c65">cb1</font> being    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c66">cb2</font> being    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font>, <font color="Maroon" title="c67">Q1</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E113:21_3_1"/><span class="lab"><font color="Green" title="E88">A138</font></span>: 
<font color="Maroon" title="c67">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c65">cb1</font>
 <span class="kw">and </span><br/><a NAME="E114:21_3_1"/><span class="lab"><font color="Green" title="E89">A139</font></span>: 
<font color="Maroon" title="c49">cc2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c65">cb1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c66">cb2</font>
 <span class="kw">and </span><br/><a NAME="E115:21_3_1"/><span class="lab"><font color="Green" title="E90">A140</font></span>: 
<span class="p1">(<span class="default"><a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c15">q</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c49">cc2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c67">Q1</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c66">cb2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom6.html#D2" target="_self" title="POLYNOM6:def.2">Def2</a></span>;<br/>
<a NAME="E116:21_3_1"/>
<font color="Maroon" title="c47">c2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c59">gb19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c62">gb199</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E101:21_3_1"><span class="lab"><font color="Green" title="E82">A132</font></span></a>, <a class="ref" href="finseq_1.html#T98" title="FINSEQ_1:th.98">FINSEQ_1:98</a></span>;<br/>
<a NAME="E117:21_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A141</font></span>: 
( <font color="Maroon" title="c65">cb1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c59">gb19</font> &amp; <font color="Maroon" title="c66">cb2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c62">gb199</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E114:21_3_1"><span class="lab"><font color="Green" title="E89">A139</font></span></a>, <a class="ref" href="polynom6.html#T7" target="_self" title="POLYNOM6:th.7">Th7</a></span>;<br/>
<a NAME="E118:21_3_1"/><span class="lab"><font color="Green" title="E92">A142</font></span>: 
<font color="Maroon" title="c55">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:21_3_1"><span class="lab"><font color="Green" title="E56">A106</font></span></a>, <a class="txt" href="#E87:21_3_1"><span class="lab"><font color="Green" title="E72">A122</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/>
<a NAME="E119:21_3_1"/>
<font color="Maroon" title="c64">b199</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c62">gb199</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E99:21_3_1"><span class="lab"><font color="Green" title="E80">A130</font></span></a>, <a class="txt" href="#E107:21_3_1"><span class="lab"><font color="Green" title="E85">A135</font></span></a>, <a class="ref" href="finseq_1.html#T98" title="FINSEQ_1:th.98">FINSEQ_1:98</a></span>;<br/>
<a NAME="E120:21_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A143</font></span>: 
<font color="Maroon" title="c57">qb19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c64">b199</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">fq</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c47">c2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E86:21_3_1"><span class="lab"><font color="Green" title="E71">A121</font></span></a>, <a class="txt" href="#E96:21_3_1"><span class="lab"><font color="Green" title="E78">A128</font></span></a>, <a class="txt" href="#E113:21_3_1"><span class="lab"><font color="Green" title="E88">A138</font></span></a>, <a class="txt" href="#E115:21_3_1"><span class="lab"><font color="Green" title="E90">A140</font></span></a>, <a class="txt" href="#E117:21_3_1"><span class="lab"><font color="Green" title="E91">A141</font></span></a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c32">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c41">k</font> = 
<font color="Maroon" title="c32">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E62:21_3_1"><span class="lab"><font color="Green" title="E55">A105</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">fp</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c46">c1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">fq</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c47">c2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E65:21_3_1"><span class="lab"><font color="Green" title="E57">A107</font></span></a>, <a class="txt" href="#E67:21_3_1"><span class="lab"><font color="Green" title="E59">A109</font></span></a>, <a class="txt" href="#E118:21_3_1"><span class="lab"><font color="Green" title="E92">A142</font></span></a>, <a class="txt" href="#E108:21_3_1"><span class="lab"><font color="Green" title="E86">A136</font></span></a>, <a class="txt" href="#E110:21_3_1"><span class="lab"><font color="Green" title="E87">A137</font></span></a>, <a class="txt" href="#E120:21_3_1"><span class="lab"><font color="Green" title="E93">A143</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</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><span class="kw">thus </span><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">y</font></span>)</span> = 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c15">q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>
<br/>.= 
<font color="Maroon" title="c18">fp</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c19">fq</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_3"><span class="lab"><font color="Green" title="E19">A18</font></span></a>, <a class="ref" href="polynom1.html#D26" title="POLYNOM1:def.26">POLYNOM1:def 26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</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>
<span class="kw">hence </span><a NAME="E9:21"/><span class="lab"><font color="Green" title="E19">A144</font></span>: 
<font color="Maroon" title="c9">f</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a></span>; <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">f</font> is  <a href="group_1.html#V6" title="GROUP_1:attr.6">unity-preserving</a>  &amp; <font color="Maroon" title="c9">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c9">f</font> is  <a href="quofield.html#V2" title="QUOFIELD:attr.2">RingEpimorphism</a>  )</span><br/>

<a NAME="E10:21"/><span class="lab"><font color="Green" title="E20">A145</font></span>: 
for <font color="Olive" title="b1">b</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span> holds  <span class="p1">(<span class="default"><a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c6">1P1</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">1P2</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b1">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">b</font> be    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</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"><a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c6">1P1</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> <font color="Maroon" title="c7">1P2</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c10">b</font></span><br/>

<a NAME="E1:21_4"/><span class="lab"><font color="Green" title="E21">A146</font></span>: 
<font color="Maroon" title="c7">1P2</font> <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"><a href="polynom1.html#K10" title="POLYNOM1:func.10">1_</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</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="ref" href="polynom1.html#T90" title="POLYNOM1:th.90">POLYNOM1:90</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">b1</font> being    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c12">b2</font> being    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font>, <font color="Maroon" title="c13">Q1</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E3:21_4"/><span class="lab"><font color="Green" title="E22">A147</font></span>: 
<font color="Maroon" title="c13">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">1P1</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c11">b1</font>
 <span class="kw">and </span><br/><a NAME="E4:21_4"/><span class="lab"><font color="Green" title="E23">A148</font></span>: 
<font color="Maroon" title="c10">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c11">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c12">b2</font>
 <span class="kw">and </span><br/><a NAME="E5:21_4"/><span class="lab"><font color="Green" title="E24">A149</font></span>: 
<span class="p1">(<span class="default"><a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c6">1P1</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> <font color="Maroon" title="c13">Q1</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c12">b2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom6.html#D2" target="_self" title="POLYNOM6:def.2">Def2</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_4_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="E1:21_4_1"/>
( <font color="Maroon" title="c10">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span> or <font color="Maroon" title="c10">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:21_4_1_1"/><span class="lab"><font color="Green" title="E25">A150</font></span>: 
<font color="Maroon" title="c10">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</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"><a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c6">1P1</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> <font color="Maroon" title="c7">1P2</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c10">b</font></span><br/><div class="add"><a NAME="E2:21_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A151</font></span>: 
<font color="Maroon" title="c11">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <font color="Maroon" title="c1">o1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_4"><span class="lab"><font color="Green" title="E23">A148</font></span></a>, <a class="ref" href="polynom6.html#T5" target="_self" title="POLYNOM6:th.5">Th5</a></span>;<br/><a NAME="E3:21_4_1_1"/><span class="lab"><font color="Green" title="E27">A152</font></span>: 
<font color="Maroon" title="c12">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <font color="Maroon" title="c2">o2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_4"><span class="lab"><font color="Green" title="E23">A148</font></span></a>, <a class="txt" href="#E1:21_4_1_1"><span class="lab"><font color="Green" title="E25">A150</font></span></a>, <a class="ref" href="polynom6.html#T5" target="_self" title="POLYNOM6:th.5">Th5</a></span>;<br/><font color="Maroon" title="c13">Q1</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">1_</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c11">b1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_4"><span class="lab"><font color="Green" title="E22">A147</font></span></a>, <a class="ref" href="polynom1.html#T90" title="POLYNOM1:th.90">POLYNOM1:90</a></span>
<br/>.= 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_4_1_1"><span class="lab"><font color="Green" title="E26">A151</font></span></a>, <a class="ref" href="polynom1.html#T84" title="POLYNOM1:th.84">POLYNOM1:84</a></span>
;<br/><span class="kw">then </span><font color="Maroon" title="c13">Q1</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c12">b2</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">1_</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c12">b2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#T90" title="POLYNOM1:th.90">POLYNOM1:90</a></span>
<br/>.= 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">L</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_4_1_1"><span class="lab"><font color="Green" title="E27">A152</font></span></a>, <a class="ref" href="polynom1.html#T84" title="POLYNOM1:th.84">POLYNOM1:84</a></span>
<br/>.= 
<font color="Maroon" title="c7">1P2</font> <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="#E1:21_4"><span class="lab"><font color="Green" title="E21">A146</font></span></a>, <a class="txt" href="#E1:21_4_1_1"><span class="lab"><font color="Green" title="E25">A150</font></span></a>, <a class="ref" href="polynom1.html#T84" title="POLYNOM1:th.84">POLYNOM1:84</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:21_4_1_1"/>
<span class="p1">(<span class="default"><a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c6">1P1</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> <font color="Maroon" title="c7">1P2</font> <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="#E5:21_4"><span class="lab"><font color="Green" title="E24">A149</font></span></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="21_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:21_4_1_2"/><span class="lab"><font color="Green" title="E25">A153</font></span>: 
<font color="Maroon" title="c10">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</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"><a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c6">1P1</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> <font color="Maroon" title="c7">1P2</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c10">b</font></span><br/><div class="add"><a NAME="E2:21_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A154</font></span>: 
( <font color="Maroon" title="c11">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <font color="Maroon" title="c1">o1</font> or <font color="Maroon" title="c12">b2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <font color="Maroon" title="c2">o2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_4"><span class="lab"><font color="Green" title="E23">A148</font></span></a>, <a class="ref" href="polynom6.html#T5" target="_self" title="POLYNOM6:th.5">Th5</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_4_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="E1:21_4_1_2_1_1"/>
( <font color="Maroon" title="c11">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <font color="Maroon" title="c1">o1</font> or <font color="Maroon" title="c11">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <font color="Maroon" title="c1">o1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_4_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:21_4_1_2_1_1_1"/><span class="lab"><font color="Green" title="E27">A155</font></span>: 
<font color="Maroon" title="c11">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <font color="Maroon" title="c1">o1</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"><a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c6">1P1</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> <font color="Maroon" title="c7">1P2</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c10">b</font></span><br/><div class="add"><font color="Maroon" title="c13">Q1</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">1_</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c11">b1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_4"><span class="lab"><font color="Green" title="E22">A147</font></span></a>, <a class="ref" href="polynom1.html#T90" title="POLYNOM1:th.90">POLYNOM1:90</a></span>
<br/>.= 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_4_1_2_1_1_1"><span class="lab"><font color="Green" title="E27">A155</font></span></a>, <a class="ref" href="polynom1.html#T84" title="POLYNOM1:th.84">POLYNOM1:84</a></span>
<br/>.= 
 <a href="polynom1.html#K10" title="POLYNOM1:func.10">1_</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#T90" title="POLYNOM1:th.90">POLYNOM1:90</a></span>
;<br/><span class="kw">then </span><font color="Maroon" title="c13">Q1</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c12">b2</font> = 
 <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="txt" href="#E2:21_4_1_2"><span class="lab"><font color="Green" title="E26">A154</font></span></a>, <a class="txt" href="#E1:21_4_1_2_1_1_1"><span class="lab"><font color="Green" title="E27">A155</font></span></a>, <a class="ref" href="polynom1.html#T84" title="POLYNOM1:th.84">POLYNOM1:84</a></span>
<br/>.= 
<font color="Maroon" title="c7">1P2</font> <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="#E1:21_4"><span class="lab"><font color="Green" title="E21">A146</font></span></a>, <a class="txt" href="#E1:21_4_1_2"><span class="lab"><font color="Green" title="E25">A153</font></span></a>, <a class="ref" href="polynom1.html#T84" title="POLYNOM1:th.84">POLYNOM1:84</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:21_4_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c6">1P1</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> <font color="Maroon" title="c7">1P2</font> <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="#E5:21_4"><span class="lab"><font color="Green" title="E24">A149</font></span></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="21_4_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:21_4_1_2_1_1_2"/><span class="lab"><font color="Green" title="E27">A156</font></span>: 
<font color="Maroon" title="c11">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <font color="Maroon" title="c1">o1</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"><a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c6">1P1</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> <font color="Maroon" title="c7">1P2</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c10">b</font></span><br/><div class="add"><font color="Maroon" title="c13">Q1</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">1_</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c11">b1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_4"><span class="lab"><font color="Green" title="E22">A147</font></span></a>, <a class="ref" href="polynom1.html#T90" title="POLYNOM1:th.90">POLYNOM1:90</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E27">A156</font></span></a>, <a class="ref" href="polynom1.html#T84" title="POLYNOM1:th.84">POLYNOM1:84</a></span>
<br/>.= 
 <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>
;<br/><span class="kw">then </span><font color="Maroon" title="c13">Q1</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c12">b2</font> = 
 <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="polynom1.html#T81" title="POLYNOM1:th.81">POLYNOM1:81</a></span>
<br/>.= 
<font color="Maroon" title="c7">1P2</font> <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="#E1:21_4"><span class="lab"><font color="Green" title="E21">A146</font></span></a>, <a class="txt" href="#E1:21_4_1_2"><span class="lab"><font color="Green" title="E25">A153</font></span></a>, <a class="ref" href="polynom1.html#T84" title="POLYNOM1:th.84">POLYNOM1:84</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:21_4_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c6">1P1</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> <font color="Maroon" title="c7">1P2</font> <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="#E5:21_4"><span class="lab"><font color="Green" title="E24">A149</font></span></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="E4:21_4_1_2"/>
<span class="p1">(<span class="default"><a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c6">1P1</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> <font color="Maroon" title="c7">1P2</font> <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>
<font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p2">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p3">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span></span>)</span> = 
 <a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c6">1P1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21"><span class="lab"><font color="Green" title="E17">A2</font></span></a></span>
<br/>.= 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:21"><span class="lab"><font color="Green" title="E20">A145</font></span></a>, <a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E12:21"/><span class="lab"><font color="Green" title="E21">A157</font></span>: 
<font color="Maroon" title="c9">f</font> is  <a href="group_1.html#V6" title="GROUP_1:attr.6">unity-preserving</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D17" title="GROUP_1:def.17">GROUP_1:def 17</a></span>; <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">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c9">f</font> is  <a href="quofield.html#V2" title="QUOFIELD:attr.2">RingEpimorphism</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E13:21"/>
<font color="Maroon" title="c9">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</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">f</font> is  <a href="quofield.html#V2" title="QUOFIELD:attr.2">RingEpimorphism</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x1</font>, <font color="Maroon" title="c11">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c9">f</font> or  not <font color="Maroon" title="c11">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c9">f</font> or  not <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font> or <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_6"/>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c9">f</font> or  not <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font> or <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x2</font> )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">x19</font> = <font color="Maroon" title="c10">x1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</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">assume </span><a NAME="E3:21_6"/>
<font color="Maroon" title="c11">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font> or <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x2</font> )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">x29</font> = <font color="Maroon" title="c11">x2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</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">reconsider </span><font color="Maroon" title="c14">x299</font> = <font color="Maroon" title="c13">x29</font> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">x199</font> = <font color="Maroon" title="c12">x19</font> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/>
<a NAME="E7:21_6"/><span class="lab"><font color="Green" title="E22">A158</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c14">x299</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21"><span class="lab"><font color="Green" title="E17">A2</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">w2</font> = <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x29</font> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font> ;<br/>
<a NAME="E9:21_6"/><span class="lab"><font color="Green" title="E23">A159</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c15">x199</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21"><span class="lab"><font color="Green" title="E17">A2</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">w1</font> = <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x19</font> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font> ;<br/>
<span class="kw">assume </span><a NAME="E11:21_6"/><span class="lab"><font color="Green" title="E24">A160</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font>
 ; <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="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x2</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_6_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">b1</font> be    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">o1</font>; <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="c15">x199</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c18">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x299</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c18">b1</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c19">x199b1</font> = <font color="Maroon" title="c15">x199</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c18">b1</font>, <font color="Maroon" title="c20">x299b1</font> = <font color="Maroon" title="c14">x299</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c18">b1</font> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_6_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c21">b2</font> be    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font>; <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="c19">x199b1</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c21">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x299b1</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c21">b2</font></span><br/><span class="kw">set </span><font color="Maroon" title="c22">b</font> = <font color="Maroon" title="c18">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c21">b2</font>;<br/><span class="kw">consider </span><font color="Maroon" title="c23">b19</font> being    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c24">b29</font> being    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font>, <font color="Maroon" title="c25">Q1</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E2:21_6_1_1"/><span class="lab"><font color="Green" title="E25">A161</font></span>: 
<font color="Maroon" title="c25">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x199</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c23">b19</font>
 <span class="kw">and </span><br/><a NAME="E3:21_6_1_1"/><span class="lab"><font color="Green" title="E26">A162</font></span>: 
<font color="Maroon" title="c18">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c21">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c23">b19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c24">b29</font>
 <span class="kw">and </span><br/><a NAME="E4:21_6_1_1"/><span class="lab"><font color="Green" title="E27">A163</font></span>: 
<font color="Maroon" title="c17">w1</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c21">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">Q1</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c24">b29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:21_6"><span class="lab"><font color="Green" title="E23">A159</font></span></a>, <a class="ref" href="polynom6.html#D2" target="_self" title="POLYNOM6:def.2">Def2</a></span>;<br/><a NAME="E5:21_6_1_1"/><span class="lab"><font color="Green" title="E28">A164</font></span>: 
( <font color="Maroon" title="c18">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c23">b19</font> &amp; <font color="Maroon" title="c21">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c24">b29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_6_1_1"><span class="lab"><font color="Green" title="E26">A162</font></span></a>, <a class="ref" href="polynom6.html#T7" target="_self" title="POLYNOM6:th.7">Th7</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c26">c1</font> being    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c27">c2</font> being    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font>, <font color="Maroon" title="c28">Q19</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E7:21_6_1_1"/><span class="lab"><font color="Green" title="E29">A165</font></span>: 
<font color="Maroon" title="c28">Q19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x299</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c26">c1</font>
 <span class="kw">and </span><br/><a NAME="E8:21_6_1_1"/><span class="lab"><font color="Green" title="E30">A166</font></span>: 
<font color="Maroon" title="c18">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c21">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c26">c1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c27">c2</font>
 <span class="kw">and </span><br/><a NAME="E9:21_6_1_1"/><span class="lab"><font color="Green" title="E31">A167</font></span>: 
<font color="Maroon" title="c16">w2</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c21">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">Q19</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c27">c2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_6"><span class="lab"><font color="Green" title="E22">A158</font></span></a>, <a class="ref" href="polynom6.html#D2" target="_self" title="POLYNOM6:def.2">Def2</a></span>;<br/><a NAME="E10:21_6_1_1"/>
<font color="Maroon" title="c18">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c26">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_6_1_1"><span class="lab"><font color="Green" title="E30">A166</font></span></a>, <a class="ref" href="polynom6.html#T7" target="_self" title="POLYNOM6:th.7">Th7</a></span>;<br/><span class="kw">hence </span><a NAME="E11:21_6_1_1"/>
<font color="Maroon" title="c19">x199b1</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c21">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x299b1</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c21">b2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:21_6"><span class="lab"><font color="Green" title="E24">A160</font></span></a>, <a class="txt" href="#E2:21_6_1_1"><span class="lab"><font color="Green" title="E25">A161</font></span></a>, <a class="txt" href="#E4:21_6_1_1"><span class="lab"><font color="Green" title="E27">A163</font></span></a>, <a class="txt" href="#E7:21_6_1_1"><span class="lab"><font color="Green" title="E29">A165</font></span></a>, <a class="txt" href="#E8:21_6_1_1"><span class="lab"><font color="Green" title="E30">A166</font></span></a>, <a class="txt" href="#E9:21_6_1_1"><span class="lab"><font color="Green" title="E31">A167</font></span></a>, <a class="txt" href="#E5:21_6_1_1"><span class="lab"><font color="Green" title="E28">A164</font></span></a>, <a class="ref" href="polynom6.html#T7" target="_self" title="POLYNOM6:th.7">Th7</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><span class="kw">hence </span><a NAME="E3:21_6_1"/>
<font color="Maroon" title="c15">x199</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c18">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x299</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c18">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</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>
<span class="kw">hence </span><a NAME="E13:21_6"/>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</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>

<span class="kw">thus </span><a NAME="E14:21"/>
<font color="Maroon" title="c9">f</font> is  <a href="quofield.html#V1" title="QUOFIELD:attr.1">RingHomomorphism</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E9:21"><span class="lab"><font color="Green" title="E19">A144</font></span></a>, <a class="txt" href="#E12:21"><span class="lab"><font color="Green" title="E21">A157</font></span></a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="quofield.html#D22" title="QUOFIELD:def.22">QUOFIELD:def 22</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <a href="relset_1.html#K2" title="RELSET_1:func.2">K807</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#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span>,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#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font></span>)</span>,<font color="Maroon" title="c9">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font></span>)</span></span><br/>

<span class="kw">thus </span><a NAME="E15:21"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="relset_1.html#K2" title="RELSET_1:func.2">K807</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#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span>,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#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font></span>)</span>,<font color="Maroon" title="c9">f</font>)</span><br/>

<span class="kw">thus </span><a NAME="E16:21"/>
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#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">f</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">b1</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">o1</font> ex <font color="Olive" title="b2">b2</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">b2</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b1</font> );<br/>
<span class="kw">let </span><font color="Maroon" title="c10">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">y</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#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font></span>)</span> or <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_7"/>
<font color="Maroon" title="c10">y</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#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font></span>)</span>
 ; <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="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">f</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">s</font> = <font color="Maroon" title="c10">y</font> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">o1</font>,  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span>] <span class="kw">means </span> ex <font color="Olive" title="b1">h</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; ( for <font color="Olive" title="b2">b2</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font><br/> for <font color="Olive" title="b3">b</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>  st <font color="Olive" title="b3">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> $1 <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">b2</font> holds <br/><font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b3">b</font> ) );<br/>
<a NAME="E3:21_7"/><span class="lab"><font color="Green" title="E22">A168</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>  ex <font color="Olive" title="b2">y</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">o1</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_7_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">o1</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c13">b1</font> being    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c14">b2</font> being    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font><span class="kw"> such that </span><br/><a NAME="E2:21_7_1"/><span class="lab"><font color="Green" title="E23">A169</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c13">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c14">b2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom6.html#T6" target="_self" title="POLYNOM6:th.6">Th6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">y</font> = <font color="Maroon" title="c13">b1</font> as    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">o1</font> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c15">y</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c15">y</font>]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c13">b1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">b2</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font> st <br/>( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b1">b2</font> &amp; <font color="Maroon" title="c15">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b1</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c14">b2</font>
; <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="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c14">b2</font> &amp; <font color="Maroon" title="c15">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b1</font> )</span><br/>

<span class="kw">thus </span><a NAME="E4:21_7_1"/>
<font color="Maroon" title="c12">x</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c13">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c14">b2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_7_1"><span class="lab"><font color="Green" title="E23">A169</font></span></a></span>; <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="c15">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b1</font></span><br/>

<span class="kw">thus </span><a NAME="E5:21_7_1"/>
<font color="Maroon" title="c15">y</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c13">b1</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>
<span class="kw">consider </span><font color="Maroon" title="c12">kk</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">o1</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E5:21_7"/><span class="lab"><font color="Green" title="E23">A170</font></span>: 
for <font color="Olive" title="b1">b</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c12">kk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E3:21_7"><span class="lab"><font color="Green" title="E22">A168</font></span></a>);<br/></span>
<a NAME="E6:21_7"/><span class="lab"><font color="Green" title="E24">A171</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">o1</font>  ex <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_7_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">b1</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">o1</font> ex <font color="Olive" title="b2">b2</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">b2</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b2</font> );<br/>
<span class="kw">let </span><font color="Maroon" title="c13">x</font> be    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">o1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span> st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c13">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c14">b1</font> = <font color="Maroon" title="c13">x</font> as    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">o1</font> ;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>5</sub></font>[   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font>,  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font>] <span class="kw">means </span>for <font color="Olive" title="b1">b</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>  st <font color="Olive" title="b1">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c14">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> $1 holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b1">b</font>;<br/>
<a NAME="E2:21_7_2"/><span class="lab"><font color="Green" title="E25">A172</font></span>: 
for <font color="Olive" title="b1">p</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font>  ex <font color="Olive" title="b2">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font> st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_7_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">p</font> be    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font> st <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c15">p</font>,<font color="Olive" title="b1">q</font>]</span><br/>

<span class="kw">take </span><font color="Maroon" title="c16">q</font> = <font color="Maroon" title="c11">s</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c15">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"> <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c15">p</font>,<font color="Maroon" title="c16">q</font>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c17">b</font> be    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>; <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="c17">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c14">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c15">p</font> implies <font color="Maroon" title="c16">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c17">b</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_7_2_1"/>
<font color="Maroon" title="c17">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c14">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c15">p</font>
 ; <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="c16">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c17">b</font></span><br/>

<span class="kw">hence </span><a NAME="E2:21_7_2_1"/>
<font color="Maroon" title="c16">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c17">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>
<span class="kw">consider </span><font color="Maroon" title="c15">t</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E4:21_7_2"/><span class="lab"><font color="Green" title="E26">A173</font></span>: 
for <font color="Olive" title="b1">b2</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font> holds  <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">b2</font>,<font color="Maroon" title="c15">t</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b1">b2</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E2:21_7_2"><span class="lab"><font color="Green" title="E25">A172</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c16">t</font> = <font color="Maroon" title="c15">t</font> as   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font> ;<br/>
<a NAME="E6:21_7_2"/><span class="lab"><font color="Green" title="E27">A174</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>  ex <font color="Olive" title="b2">y</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font> st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_7_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">x</font> be    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font> st <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c17">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c18">b1</font> being    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c19">b2</font> being    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font><span class="kw"> such that </span><br/><a NAME="E2:21_7_2_2"/><span class="lab"><font color="Green" title="E28">A175</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c18">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c19">b2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom6.html#T6" target="_self" title="POLYNOM6:th.6">Th6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">y</font> = <font color="Maroon" title="c19">b2</font> as    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c20">y</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c20">y</font>]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c18">b1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">b2</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font> st <br/>( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b1">b2</font> &amp; <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b2</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c19">b2</font>
; <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="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c19">b2</font> &amp; <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b2</font> )</span><br/>

<span class="kw">thus </span><a NAME="E4:21_7_2_2"/>
<font color="Maroon" title="c17">x</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c18">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c19">b2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_7_2_2"><span class="lab"><font color="Green" title="E28">A175</font></span></a></span>; <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="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b2</font></span><br/>

<span class="kw">thus </span><a NAME="E5:21_7_2_2"/>
<font color="Maroon" title="c20">y</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c19">b2</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>
<span class="kw">consider </span><font color="Maroon" title="c17">kk</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E8:21_7_2"/><span class="lab"><font color="Green" title="E28">A176</font></span>: 
for <font color="Olive" title="b1">b</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span> holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c17">kk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E6:21_7_2"><span class="lab"><font color="Green" title="E27">A174</font></span></a>);<br/></span>
<a NAME="E9:21_7_2"/>
 <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c16">t</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">kk</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c11">s</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_7_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c16">t</font> or <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">kk</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c11">s</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_7_2_3"/><span class="lab"><font color="Green" title="E29">A177</font></span>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c16">t</font>
 ; <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="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">kk</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c11">s</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">b2</font> = <font color="Maroon" title="c18">x</font> as    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c20">b</font> = <font color="Maroon" title="c14">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c19">b2</font>;<br/>
<a NAME="E3:21_7_2_3"/>
<font color="Maroon" title="c16">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="txt" href="#E1:21_7_2_3"><span class="lab"><font color="Green" title="E29">A177</font></span></a>, <a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></span>;<br/>
<a NAME="E4:21_7_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c11">s</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c19">b2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="txt" href="#E4:21_7_2"><span class="lab"><font color="Green" title="E26">A173</font></span></a></span>;<br/>
<a NAME="E5:21_7_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A178</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c17">kk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span> &amp; <font color="Maroon" title="c14">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c19">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c11">s</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>, <a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></span>;<br/>
<a NAME="E6:21_7_2_3"/>
 ex <font color="Olive" title="b1">b19</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">o1</font> ex <font color="Olive" title="b2">b29</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font> st <br/>( <font color="Maroon" title="c14">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c19">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">b19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">b29</font> &amp; <font color="Maroon" title="c17">kk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c19">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b29</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_7_2"><span class="lab"><font color="Green" title="E28">A176</font></span></a></span>;<br/>
<a NAME="E7:21_7_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">kk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c19">b2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom6.html#T7" target="_self" title="POLYNOM6:th.7">Th7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:21_7_2_3"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">kk</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c11">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21_7_2_3"><span class="lab"><font color="Green" title="E30">A178</font></span></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</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="E10:21_7_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">t</font> is   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D10" title="POLYNOM1:def.10">POLYNOM1:def 10</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">t99</font> = <font color="Maroon" title="c16">t</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">t9</font> = <font color="Maroon" title="c16">t</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c18">t99</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c18">t99</font>]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c19">t9</font>
; <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="c19">t9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">t99</font> &amp; ( for <font color="Olive" title="b1">b2</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font><br/> for <font color="Olive" title="b2">b</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>  st <font color="Olive" title="b2">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c13">x</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b1">b2</font> holds <br/><font color="Maroon" title="c19">t9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b2">b</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E13:21_7_2"/>
<font color="Maroon" title="c18">t99</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">t9</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">b2</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font><br/> for <font color="Olive" title="b2">b</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>  st <font color="Olive" title="b2">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c13">x</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b1">b2</font> holds <br/><font color="Maroon" title="c19">t9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b2">b</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c20">b2</font> be    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</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">b</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>  st <font color="Olive" title="b1">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c13">x</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c20">b2</font> holds <br/><font color="Maroon" title="c19">t9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b1">b</font></span><br/><span class="kw">let </span><font color="Maroon" title="c21">b</font> be    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>; <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="c21">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c13">x</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c20">b2</font> implies <font color="Maroon" title="c19">t9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c21">b</font> )</span><br/>



<span class="kw">assume </span><a NAME="E14:21_7_2"/>
<font color="Maroon" title="c21">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c13">x</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c20">b2</font>
 ; <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="c19">t9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c21">b</font></span><br/>

<span class="kw">hence </span><a NAME="E15:21_7_2"/>
<font color="Maroon" title="c19">t9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c21">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_7_2"><span class="lab"><font color="Green" title="E26">A173</font></span></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>
<span class="kw">consider </span><font color="Maroon" title="c13">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">o1</font></span>)</span>,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#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E8:21_7"/><span class="lab"><font color="Green" title="E25">A179</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">o1</font> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c13">g</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E6:21_7"><span class="lab"><font color="Green" title="E24">A171</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c14">g</font> = <font color="Maroon" title="c13">g</font> as   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">o1</font></span>)</span>,<span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span> ;<br/>
<a NAME="E10:21_7"/><span class="lab"><font color="Green" title="E26">A180</font></span>: 
 <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c14">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">kk</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c11">s</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_7_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c14">g</font> or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">kk</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c11">s</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_7_3"/><span class="lab"><font color="Green" title="E27">A181</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c14">g</font>
 ; <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="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">kk</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c11">s</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">b1</font> = <font color="Maroon" title="c15">x</font> as    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">o1</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c17">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E4:21_7_3"/><span class="lab"><font color="Green" title="E28">A182</font></span>: 
<font color="Maroon" title="c17">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c16">b1</font>
 <span class="kw">and </span><br/><a NAME="E5:21_7_3"/><span class="lab"><font color="Green" title="E29">A183</font></span>: 
for <font color="Olive" title="b1">b2</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font><br/> for <font color="Olive" title="b2">b</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>  st <font color="Olive" title="b2">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c16">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b1">b2</font> holds <br/><font color="Maroon" title="c17">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b2">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_7"><span class="lab"><font color="Green" title="E25">A179</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">h</font> = <font color="Maroon" title="c17">h</font> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_7_3"><span class="lab"><font color="Green" title="E28">A182</font></span></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/>
<a NAME="E7:21_7_3"/>
<font color="Maroon" title="c14">g</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c16">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_7_3"><span class="lab"><font color="Green" title="E27">A181</font></span></a>, <a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></span>;<br/>
<a NAME="E8:21_7_3"/><span class="kw">then </span>
<font color="Maroon" title="c14">g</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c16">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">b2</font> being    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font><span class="kw"> such that </span><br/><a NAME="E10:21_7_3"/><span class="lab"><font color="Green" title="E30">A184</font></span>: 
<font color="Maroon" title="c19">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c18">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_7_3"><span class="lab"><font color="Green" title="E28">A182</font></span></a>, <a class="ref" href="polynom2.html#T19" title="POLYNOM2:th.19">POLYNOM2:19</a>, <a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c20">b</font> = <font color="Maroon" title="c16">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c19">b2</font>;<br/>
<a NAME="E11:21_7_3"/>
<font color="Maroon" title="c18">h</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c19">b2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="txt" href="#E10:21_7_3"><span class="lab"><font color="Green" title="E30">A184</font></span></a>, <a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></span>;<br/>
<a NAME="E12:21_7_3"/><span class="kw">then </span>
<font color="Maroon" title="c11">s</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c19">b2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="txt" href="#E5:21_7_3"><span class="lab"><font color="Green" title="E29">A183</font></span></a></span>;<br/>
<a NAME="E13:21_7_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A185</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">kk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span> &amp; <font color="Maroon" title="c16">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c19">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c11">s</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>, <a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></span>;<br/>
<a NAME="E14:21_7_3"/>
 ex <font color="Olive" title="b1">b19</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">o1</font> ex <font color="Olive" title="b2">b29</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font> st <br/>( <font color="Maroon" title="c16">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c19">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">b19</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">b29</font> &amp; <font color="Maroon" title="c12">kk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c19">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b19</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21_7"><span class="lab"><font color="Green" title="E23">A170</font></span></a></span>;<br/>
<a NAME="E15:21_7_3"/><span class="kw">then </span>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">kk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c19">b2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom6.html#T7" target="_self" title="POLYNOM6:th.7">Th7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:21_7_3"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">kk</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c11">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21_7_3"><span class="lab"><font color="Green" title="E31">A185</font></span></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</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="E11:21_7"/><span class="kw">then </span>
<font color="Maroon" title="c14">g</font> is   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D10" title="POLYNOM1:def.10">POLYNOM1:def 10</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">g</font> = <font color="Maroon" title="c14">g</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">g9</font> = <font color="Maroon" title="c15">g</font> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:21_7"><span class="lab"><font color="Green" title="E26">A180</font></span></a>, <a class="ref" href="polynom1.html#D10" title="POLYNOM1:def.10">POLYNOM1:def 10</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_7_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">b</font> be    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>; <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="c11">s</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c17">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c16">g9</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c17">b</font></span><br/><span class="kw">consider </span><font color="Maroon" title="c18">b1</font> being    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c19">b2</font> being    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font>, <font color="Maroon" title="c20">Q1</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E2:21_7_4"/><span class="lab"><font color="Green" title="E27">A186</font></span>: 
( <font color="Maroon" title="c20">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g9</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c18">b1</font> &amp; <font color="Maroon" title="c17">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c18">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c19">b2</font> &amp; <span class="p1">(<span class="default"><a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c16">g9</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c17">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">Q1</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c19">b2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom6.html#D2" target="_self" title="POLYNOM6:def.2">Def2</a></span>;<br/><a NAME="E3:21_7_4"/>
 ex <font color="Olive" title="b1">h</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g9</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c18">b1</font> &amp; ( for <font color="Olive" title="b2">b2</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">o2</font><br/> for <font color="Olive" title="b3">b</font> being   <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>  st <font color="Olive" title="b3">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c18">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">b2</font> holds <br/><font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b3">b</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_7"><span class="lab"><font color="Green" title="E25">A179</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:21_7_4"/>
<font color="Maroon" title="c11">s</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c17">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c16">g9</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c17">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_7_4"><span class="lab"><font color="Green" title="E27">A186</font></span></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>
<span class="kw">then </span><span class="lab"><font color="Green" title="E27">A187</font></span>: <font color="Maroon" title="c10">y</font> = 
 <a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c16">g9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></span>
<br/>.= 
<font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">g</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21"><span class="lab"><font color="Green" title="E17">A2</font></span></a></span>
;<br/>
<a NAME="E16:21_7"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</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="E17:21_7"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:21_7"><span class="lab"><font color="Green" title="E27">A187</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</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>
