<?xml version="1.0"?>
<div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c3">Z</font> = <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font></span>)</span> as   <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">n</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font> ;<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> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; <font color="Olive" title="b3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $3 &amp; <font color="Olive" title="b1">p</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Olive" title="b2">q</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b3">r</font> );<br/>
<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> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; <font color="Olive" title="b3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $3 &amp; <font color="Olive" title="b1">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Olive" title="b2">q</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b3">r</font> );<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">x9</font> being   <a href="polynom1.html#V1" title="POLYNOM1:attr.1">finite-Support</a>  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">Series</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>;<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> ] <span class="kw">means </span> ex <font color="Olive" title="b1">x9</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">Series</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b1">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b1">x9</font> is  <a href="polynom1.html#V1" title="POLYNOM1:attr.1">finite-Support</a>  );<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">P</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:116_1_1"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> iff <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>] )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="subset_1.html#S3" title="SUBSET_1:sch.3">SUBSET_1:sch 3</a>();<br/></span>
<a NAME="E5:116_1_1"/>
<font color="Maroon" title="c4">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">P</font> = <font color="Maroon" title="c5">P</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:116_1_1"><span class="lab"><font color="Green" title="E32">A1</font></span></a></span>;<br/>
<div><span class="lab"><font color="Green" title="E33">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="116_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c6">P</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">u</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c6">P</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>,<font color="Olive" title="b1">u</font>]</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c9">p</font> = <font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c10">q</font> = <font color="Maroon" title="c8">y</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font> ;<br/><span class="kw">consider </span><font color="Maroon" title="c11">p9</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">Series</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font><span class="kw"> such that </span><br/><a NAME="E3:116_1_1_1"/><span class="lab"><font color="Green" title="E34">A3</font></span>: 
<font color="Maroon" title="c11">p9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c9">p</font>
 <span class="kw">and </span><br/><a NAME="E4:116_1_1_1"/><span class="lab"><font color="Green" title="E35">A4</font></span>: 
<font color="Maroon" title="c11">p9</font> is  <a href="polynom1.html#V1" title="POLYNOM1:attr.1">finite-Support</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:116_1_1"><span class="lab"><font color="Green" title="E32">A1</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">q9</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">Series</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font><span class="kw"> such that </span><br/><a NAME="E6:116_1_1_1"/><span class="lab"><font color="Green" title="E36">A5</font></span>: 
<font color="Maroon" title="c12">q9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c10">q</font>
 <span class="kw">and </span><br/><a NAME="E7:116_1_1_1"/><span class="lab"><font color="Green" title="E37">A6</font></span>: 
<font color="Maroon" title="c12">q9</font> is  <a href="polynom1.html#V1" title="POLYNOM1:attr.1">finite-Support</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:116_1_1"><span class="lab"><font color="Green" title="E32">A1</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">p9</font> = <font color="Maroon" title="c11">p9</font>, <font color="Maroon" title="c14">q9</font> = <font color="Maroon" title="c12">q9</font> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:116_1_1_1"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E7:116_1_1_1"><span class="lab"><font color="Green" title="E37">A6</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c15">r</font> = <font color="Maroon" title="c13">p9</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c14">q9</font>;<br/><a NAME="E9:116_1_1_1"/>
<font color="Maroon" title="c13">p9</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c14">q9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">u</font> = <font color="Maroon" title="c13">p9</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c14">q9</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c6">P</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:116_1_1"><span class="lab"><font color="Green" title="E32">A1</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c17">u</font> = <font color="Maroon" title="c16">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>2</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c17">u</font>]</span><br/><span class="kw">thus </span><a NAME="E11:116_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c17">u</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:116_1_1_1"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E6:116_1_1_1"><span class="lab"><font color="Green" title="E36">A5</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="c7">fadd</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c6">P</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c6">P</font><span class="kw"> such that </span><br/><a NAME="E9:116_1_1"/><span class="lab"><font color="Green" title="E34">A7</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c6">P</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Maroon" title="c7">fadd</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="binop_1.html#S3" title="BINOP_1:sch.3">BINOP_1:sch 3</a>(<a class="txt" href="#E7:116_1_1"><span class="lab"><font color="Green" title="E33">A2</font></span></a>);<br/></span>
<div><span class="lab"><font color="Green" title="E35">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="116_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">y</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c6">P</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">u</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c6">P</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font>,<font color="Olive" title="b1">u</font>]</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c10">p</font> = <font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c11">q</font> = <font color="Maroon" title="c9">y</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font> ;<br/><span class="kw">consider </span><font color="Maroon" title="c12">p9</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">Series</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font><span class="kw"> such that </span><br/><a NAME="E3:116_1_1_2"/><span class="lab"><font color="Green" title="E36">A9</font></span>: 
<font color="Maroon" title="c12">p9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c10">p</font>
 <span class="kw">and </span><br/><a NAME="E4:116_1_1_2"/><span class="lab"><font color="Green" title="E37">A10</font></span>: 
<font color="Maroon" title="c12">p9</font> is  <a href="polynom1.html#V1" title="POLYNOM1:attr.1">finite-Support</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:116_1_1"><span class="lab"><font color="Green" title="E32">A1</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c13">q9</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">Series</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font><span class="kw"> such that </span><br/><a NAME="E6:116_1_1_2"/><span class="lab"><font color="Green" title="E38">A11</font></span>: 
<font color="Maroon" title="c13">q9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c11">q</font>
 <span class="kw">and </span><br/><a NAME="E7:116_1_1_2"/><span class="lab"><font color="Green" title="E39">A12</font></span>: 
<font color="Maroon" title="c13">q9</font> is  <a href="polynom1.html#V1" title="POLYNOM1:attr.1">finite-Support</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:116_1_1"><span class="lab"><font color="Green" title="E32">A1</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">p9</font> = <font color="Maroon" title="c12">p9</font>, <font color="Maroon" title="c15">q9</font> = <font color="Maroon" title="c13">q9</font> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:116_1_1_2"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E7:116_1_1_2"><span class="lab"><font color="Green" title="E39">A12</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c16">r</font> = <font color="Maroon" title="c14">p9</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c15">q9</font>;<br/><a NAME="E9:116_1_1_2"/>
<font color="Maroon" title="c14">p9</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c15">q9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">u</font> = <font color="Maroon" title="c14">p9</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c15">q9</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c6">P</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:116_1_1"><span class="lab"><font color="Green" title="E32">A1</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c18">u</font> = <font color="Maroon" title="c17">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="c9">y</font>,<font color="Maroon" title="c18">u</font>]</span><br/><span class="kw">thus </span><a NAME="E11:116_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c18">u</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:116_1_1_2"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="txt" href="#E6:116_1_1_2"><span class="lab"><font color="Green" title="E38">A11</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="c8">fmult</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c6">P</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c6">P</font><span class="kw"> such that </span><br/><a NAME="E12:116_1_1"/><span class="lab"><font color="Green" title="E36">A13</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c6">P</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Maroon" title="c8">fmult</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="binop_1.html#S3" title="BINOP_1:sch.3">BINOP_1:sch 3</a>(<a class="txt" href="#E10:116_1_1"><span class="lab"><font color="Green" title="E35">A8</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c9">ZZ</font> = <font color="Maroon" title="c3">Z</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">Z9</font> = <font color="Maroon" title="c3">Z</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">n</font></span>)</span>,<font color="Maroon" title="c2">L</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">Z9</font> = <font color="Maroon" title="c10">Z9</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">Series</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c12">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><a NAME="E2:116_1_1_3"/><span class="lab"><font color="Green" title="E37">A14</font></span>: 
<font color="Maroon" title="c12">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="c11">Z9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c13">x</font> = <font color="Maroon" title="c12">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">n</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:116_1_1_3"><span class="lab"><font color="Green" title="E37">A14</font></span></a></span>;<br/><a NAME="E4:116_1_1_3"/>
<font color="Maroon" title="c11">Z9</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E5:116_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:116_1_1_3"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="ref" href="polynom1.html#D9" target="_self" title="POLYNOM1:def.9">Def9</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="E17:116_1_1"/><span class="kw">then </span>
 <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c11">Z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E18:116_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">Z9</font> is  <a href="polynom1.html#V1" title="POLYNOM1:attr.1">finite-Support</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D10" target="_self" title="POLYNOM1:def.10">Def10</a></span>;<br/>
<a NAME="E19:116_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">ZZ</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:116_1_1"><span class="lab"><font color="Green" title="E32">A1</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">Ze</font> = <font color="Maroon" title="c3">Z</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c6">P</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">O</font> = <font color="Maroon" title="c3">Z</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">L</font></span>)</span> as   <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">n</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">O9</font> = <font color="Maroon" title="c13">O</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">n</font></span>)</span>,<font color="Maroon" title="c2">L</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">O9</font> = <font color="Maroon" title="c14">O9</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">Series</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">O</font> = <font color="Maroon" title="c13">O</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_1_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">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="c17">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="c15">O9</font> implies not <font color="Maroon" title="c17">x</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">n</font> ) &amp; ( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c17">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="c15">O9</font> ) )</span><br/><div><span class="kw">hereby </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">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c17">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="c15">O9</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:116_1_1_4_1"/><span class="lab"><font color="Green" title="E37">A15</font></span>: 
<font color="Maroon" title="c17">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="c15">O9</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="c17">x</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">n</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">x9</font> = <font color="Maroon" title="c17">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">n</font> ;<br/><span class="kw">assume </span><a NAME="E3:116_1_1_4_1"/>
<font color="Maroon" title="c17">x</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">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><font color="Maroon" title="c15">O9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> = 
<font color="Maroon" title="c3">Z</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c18">x9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:116_1_1_4_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:116_1_1_4_1"><span class="lab"><font color="Green" title="E37">A15</font></span></a>, <a class="ref" href="polynom1.html#D9" target="_self" title="POLYNOM1:def.9">Def9</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">assume </span><a NAME="E2:116_1_1_4"/><span class="lab"><font color="Green" title="E37">A16</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">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="c15">O9</font></span><br/><a NAME="E3:116_1_1_4"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">Z</font> <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="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></span>;<br/><a NAME="E4:116_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c15">O9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">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="c2">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:116_1_1_4"><span class="lab"><font color="Green" title="E37">A16</font></span></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></span>;<br/><span class="kw">hence </span><a NAME="E5:116_1_1_4"/>
<font color="Maroon" title="c17">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="c15">O9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:116_1_1_4"><span class="lab"><font color="Green" title="E37">A16</font></span></a>, <a class="ref" href="polynom1.html#D9" target="_self" title="POLYNOM1:def.9">Def9</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="E26:116_1_1"/><span class="kw">then </span>
 <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c15">O9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E27:116_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">O9</font> is  <a href="polynom1.html#V1" title="POLYNOM1:attr.1">finite-Support</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D10" target="_self" title="POLYNOM1:def.10">Def10</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">O</font> = <font color="Maroon" title="c16">O</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c6">P</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:116_1_1"><span class="lab"><font color="Green" title="E32">A1</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">R</font> =  <a href="algstr_0.html#G6" title="ALGSTR_0:aggr.6">doubleLoopStr</a>(# <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c7">fadd</font>,<font color="Maroon" title="c8">fmult</font>,<font color="Maroon" title="c17">O</font>,<font color="Maroon" title="c12">Ze</font> #) as  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#V36" title="ALGSTR_0:attr.36">strict</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a>  ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c18">R</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">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c18">R</font> iff <font color="Olive" title="b1">x</font> is   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> ) ) &amp; ( for <font color="Olive" title="b1">x</font>, <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="c18">R</font><br/> for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">q</font> holds <br/><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Olive" title="b4">q</font> ) &amp; ( for <font color="Olive" title="b1">x</font>, <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="c18">R</font><br/> for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">q</font> holds <br/><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Olive" title="b4">q</font> ) &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c18">R</font> <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="c1">n</font>,<font color="Maroon" title="c2">L</font> &amp;  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c18">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">1_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> )</span><br/>

<span class="kw">thus </span><a NAME="E30:116_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c18">R</font> iff <font color="Olive" title="b1">x</font> is   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">x</font>, <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="c18">R</font><br/> for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">q</font> holds <br/><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Olive" title="b4">q</font> ) &amp; ( for <font color="Olive" title="b1">x</font>, <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="c18">R</font><br/> for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">q</font> holds <br/><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Olive" title="b4">q</font> ) &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c18">R</font> <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="c1">n</font>,<font color="Maroon" title="c2">L</font> &amp;  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c18">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">1_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_1_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">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="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c18">R</font> iff <font color="Maroon" title="c19">x</font> is   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> )</span><br/>

<div><span class="kw">hereby </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="c19">x</font> is   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> implies <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c18">R</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:116_1_1_5_1"/><span class="lab"><font color="Green" title="E37">A17</font></span>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c18">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="c19">x</font> is   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">xx</font> = <font color="Maroon" title="c19">x</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font> ;<br/><a NAME="E3:116_1_1_5_1"/>
 ex <font color="Olive" title="b1">x9</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">Series</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b1">x9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c20">xx</font> &amp; <font color="Olive" title="b1">x9</font> is  <a href="polynom1.html#V1" title="POLYNOM1:attr.1">finite-Support</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:116_1_1"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E1:116_1_1_5_1"><span class="lab"><font color="Green" title="E37">A17</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:116_1_1_5_1"/>
<font color="Maroon" title="c19">x</font> is   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</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">assume </span><a NAME="E2:116_1_1_5"/><span class="lab"><font color="Green" title="E37">A18</font></span>: 
<font color="Maroon" title="c19">x</font> is   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">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="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c18">R</font></span><br/>

<a NAME="E3:116_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c19">x</font> is    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:116_1_1_5"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c18">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:116_1_1"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E2:116_1_1_5"><span class="lab"><font color="Green" title="E37">A18</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">hereby </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">x</font>, <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="c18">R</font><br/> for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">q</font> holds <br/><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Olive" title="b4">q</font> ) &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c18">R</font> <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="c1">n</font>,<font color="Maroon" title="c2">L</font> &amp;  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c18">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">1_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> )</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c19">x</font>, <font color="Maroon" title="c20">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c18">R</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">p</font>, <font color="Olive" title="b2">q</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>  st <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp; <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font> holds <br/><font color="Maroon" title="c19">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Olive" title="b2">q</font></span><br/><span class="kw">let </span><font color="Maroon" title="c21">p</font>, <font color="Maroon" title="c22">q</font> be   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">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="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">p</font> &amp; <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">q</font> implies <font color="Maroon" title="c19">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c22">q</font> )</span><br/><span class="kw">assume </span><a NAME="E1:116_1_1_6"/><span class="lab"><font color="Green" title="E37">A19</font></span>: 
( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">p</font> &amp; <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">q</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">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c22">q</font></span><br/><a NAME="E2:116_1_1_6"/>
 ex <font color="Olive" title="b1">p9</font>, <font color="Olive" title="b2">q9</font>, <font color="Olive" title="b3">r9</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b1">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x</font> &amp; <font color="Olive" title="b2">q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">y</font> &amp; <font color="Olive" title="b3">r9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fadd</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c19">x</font>,<font color="Maroon" title="c20">y</font> &amp; <font color="Olive" title="b1">p9</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Olive" title="b2">q9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b3">r9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:116_1_1"><span class="lab"><font color="Green" title="E34">A7</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:116_1_1_6"/>
<font color="Maroon" title="c19">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c22">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:116_1_1_6"><span class="lab"><font color="Green" title="E37">A19</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">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c18">R</font> <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="c1">n</font>,<font color="Maroon" title="c2">L</font> &amp;  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c18">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">1_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> )</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c19">x</font>, <font color="Maroon" title="c20">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c18">R</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">p</font>, <font color="Olive" title="b2">q</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>  st <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp; <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font> holds <br/><font color="Maroon" title="c19">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Olive" title="b2">q</font></span><br/><span class="kw">let </span><font color="Maroon" title="c21">p</font>, <font color="Maroon" title="c22">q</font> be   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">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="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">p</font> &amp; <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">q</font> implies <font color="Maroon" title="c19">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">p</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c22">q</font> )</span><br/><span class="kw">assume </span><a NAME="E1:116_1_1_7"/><span class="lab"><font color="Green" title="E37">A20</font></span>: 
( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">p</font> &amp; <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">q</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">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">p</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c22">q</font></span><br/><a NAME="E2:116_1_1_7"/>
 ex <font color="Olive" title="b1">p9</font>, <font color="Olive" title="b2">q9</font>, <font color="Olive" title="b3">r9</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b1">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x</font> &amp; <font color="Olive" title="b2">q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">y</font> &amp; <font color="Olive" title="b3">r9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">fmult</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c19">x</font>,<font color="Maroon" title="c20">y</font> &amp; <font color="Olive" title="b1">p9</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Olive" title="b2">q9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b3">r9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:116_1_1"><span class="lab"><font color="Green" title="E36">A13</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:116_1_1_7"/>
<font color="Maroon" title="c19">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">p</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c22">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:116_1_1_7"><span class="lab"><font color="Green" title="E37">A20</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">thus </span><a NAME="E33:116_1_1"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c18">R</font> <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="c1">n</font>,<font color="Maroon" title="c2">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c18">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">1_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font></span><br/>

<span class="kw">thus </span><a NAME="E34:116_1_1"/>
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c18">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">1_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</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>
