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<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">L</font> being non <a href="struct_0.html#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#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="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> <br/> for <font color="Olive" title="b2">m</font> being  <a href="polynom7.html#NM1" title="POLYNOM7:NM.1">Monomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font><br/> for <font color="Olive" title="b3">x</font> being  <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font> holds   <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K3" title="POLYNOM7:func.3">coefficient</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Olive" title="b2">m</font></span>)</span>,<font color="Olive" title="b3">x</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">L</font> be  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#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="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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">m</font> being  <a href="polynom7.html#NM1" title="POLYNOM7:NM.1">Monomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font><br/> for <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> holds   <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K3" title="POLYNOM7:func.3">coefficient</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Olive" title="b1">m</font></span>)</span>,<font color="Olive" title="b2">x</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">m</font> be   <a href="polynom7.html#NM1" title="POLYNOM7:NM.1">Monomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> holds   <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K3" title="POLYNOM7:func.3">coefficient</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c3">m</font></span>)</span>,<font color="Olive" title="b1">x</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K3" title="POLYNOM7:func.3">coefficient</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c4">x</font></span>)</span></span><br/>







<b>consider </b><font color="Maroon" title="c5">y</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="c2">L</font><b> such that </b><br/><a NAME="E2:28"/><i><font color="Green" title="E13">A1</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">y</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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> &amp;  <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">x</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="c5">y</font> &amp; ( 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 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</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="c5">y</font> holds <br/><font color="Maroon" title="c5">y</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="polynom2.html#K4" title="POLYNOM2:func.4">@</a> </span>)</span>,<font color="Maroon" title="c4">x</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="polynom2.html#D4" title="POLYNOM2:def.4">POLYNOM2:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">b</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E4:28"/><i><font color="Green" title="E14">A2</font></i>: 
for <font color="Olive" title="b1">b'</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b1">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Maroon" title="c3">m</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Olive" title="b1">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
 <b>by </b><i><a class="ref" href="polynom7.html#D4" target="_self" title="POLYNOM7:def.4">Def4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1_1"/>
( <font color="Maroon" title="c3">m</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c6">b</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> or <font color="Maroon" title="c3">m</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1"><b>case </b></a><a NAME="E1:28_1_1_1"/><i><font color="Green" title="E15">A3</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c6">b</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K3" title="POLYNOM7:func.3">coefficient</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c4">x</font></span>)</span></span><br/><div class="add"><a NAME="E2:28_1_1_1"/><i><font color="Green" title="E16">A4</font></i>: 
for <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">m</font> holds <br/><font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">m</font> implies <font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<b>assume </b><a NAME="E1:28_1_1_1_1"/><i><font color="Green" title="E17">A5</font></i>: 
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<b>assume </b><a NAME="E2:28_1_1_1_1"/><i><font color="Green" title="E18">A6</font></i>: 
not <font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>reconsider </b><font color="Maroon" title="c8">u</font> = <font color="Maroon" title="c7">u</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="txt" href="#E1:28_1_1_1_1"><i><font color="Green" title="E17">A5</font></i></a></i>;<br/>
<a NAME="E4:28_1_1_1_1"/>
<font color="Maroon" title="c8">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font>
 
<b>by </b><i><a class="txt" href="#E2:28_1_1_1_1"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:28_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">m</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c8">u</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>
 
<b>by </b><i><a class="txt" href="#E4:28"><i><font color="Green" title="E14">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:28_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:28_1_1_1_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E3:28_1_1_1"/>
for <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/><font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">m</font> )</span><br/>

<b>assume </b><a NAME="E1:28_1_1_1_2"/><i><font color="Green" title="E17">A7</font></i>: 
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">m</font></span><br/>

<a NAME="E2:28_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">u</font> = <font color="Maroon" title="c7">u</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/>
<a NAME="E4:28_1_1_1_2"/>
<font color="Maroon" title="c3">m</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c8">u</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>
 
<b>by </b><i><a class="txt" href="#E1:28_1_1_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="#E1:28_1_1_1_2"><i><font color="Green" title="E17">A7</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:28_1_1_1_2"/>
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">m</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E4:28_1_1_1"/><b>then </b><i><font color="Green" title="E17">A8</font></i>: 
 <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">m</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"><font color="Maroon" title="c6">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E2:28_1_1_1"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">sm</font> =  <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">m</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/><b>set </b><font color="Maroon" title="c8">sg</font> =  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font>;<br/><a NAME="E6:28_1_1_1"/><i><font color="Green" title="E18">A9</font></i>: 
 <a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a> <font color="Maroon" title="c7">sm</font>
 <b>by </b><i><a class="ref" href="polynom2.html#T20" title="POLYNOM2:th.20">POLYNOM2:20</a></i>;<br/><a NAME="E7:28_1_1_1"/><b>then </b><i><font color="Green" title="E19">A10</font></i>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span> <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"><font color="Maroon" title="c6">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; ( for <font color="Olive" title="b1">l</font>, <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="b1">l</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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span> &amp; <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> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span> &amp; <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> holds <br/>( <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">m</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">l</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">m</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font> ) ) )
 <b>by </b><i><a class="txt" href="#E4:28_1_1_1"><i><font color="Green" title="E17">A8</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>;<br/><a NAME="E8:28_1_1_1"/><b>then </b><i><font color="Green" title="E20">A11</font></i>: 
<font color="Maroon" title="c6">b</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"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E9:28_1_1_1"/><b>then </b><i><font color="Green" title="E21">A12</font></i>: 
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> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T33" title="FINSEQ_3:th.33">FINSEQ_3:33</a></i>;<br/><a NAME="E10:28_1_1_1"/><b>then </b><i><font color="Green" title="E22">A13</font></i>: 
for <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds <br/><font color="Olive" title="b1">u</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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E11:28_1_1_1"/>
for <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u</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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span> holds <br/><font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">u</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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span> implies <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/>

<b>assume </b><a NAME="E1:28_1_1_1_3"/><i><font color="Green" title="E23">A14</font></i>: 
<font color="Maroon" title="c9">u</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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/>

<b>assume </b><a NAME="E2:28_1_1_1_3"/><i><font color="Green" title="E24">A15</font></i>: 
not <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>reconsider </b><font color="Maroon" title="c10">u</font> = <font color="Maroon" title="c9">u</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E1:28_1_1_1_3"><i><font color="Green" title="E23">A14</font></i></a></i>;<br/>
<a NAME="E4:28_1_1_1_3"/><i><font color="Green" title="E25">A16</font></i>: 
<font color="Maroon" title="c10">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<b>by </b><i><a class="txt" href="#E2:28_1_1_1_3"><i><font color="Green" title="E24">A15</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:28_1_1_1_3"/><i><font color="Green" title="E26">A17</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1_3_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c11">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E2:28_1_1_1_3_1"/><i><font color="Green" title="E27">A18</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span> <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="Maroon" title="c11">k</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E3:28_1_1_1_3_1"/>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">l</font> where <font color="Olive" title="b1">l</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">l</font> &amp; <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D1" title="FINSEQ_1:def.1">FINSEQ_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">m'</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> <b> such that </b><br/><a NAME="E5:28_1_1_1_3_1"/><i><font color="Green" title="E28">A19</font></i>: 
( <font color="Maroon" title="c12">m'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">u</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m'</font> &amp; <font color="Maroon" title="c12">m'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font> )
 <b>by </b><i><a class="txt" href="#E1:28_1_1_1_3"><i><font color="Green" title="E23">A14</font></i></a>, <a class="txt" href="#E2:28_1_1_1_3_1"><i><font color="Green" title="E27">A18</font></i></a></i>;<br/>
<b>thus </b><a NAME="E6:28_1_1_1_3_1"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">u</font>
 <b>by </b><i><a class="txt" href="#E4:28_1_1_1_3"><i><font color="Green" title="E25">A16</font></i></a>, <a class="txt" href="#E5:28_1_1_1_3_1"><i><font color="Green" title="E28">A19</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E6:28_1_1_1_3"/>
( <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">u</font> )
 
<b>by </b><i><a class="txt" href="#E8:28_1_1_1"><i><font color="Green" title="E20">A11</font></i></a>, <a class="txt" href="#E1:28_1_1_1_3"><i><font color="Green" title="E23">A14</font></i></a>, <a class="ref" href="finseq_3.html#T33" title="FINSEQ_3:th.33">FINSEQ_3:33</a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E7:28_1_1_1_3"/><b>then </b><i><font color="Green" title="E27">A20</font></i>: 
( <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <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"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">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> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E9:28_1_1_1"><i><font color="Green" title="E21">A12</font></i></a>, <a class="txt" href="#E1:28_1_1_1_3"><i><font color="Green" title="E23">A14</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<font color="Maroon" title="c6">b</font>
<b>by </b><i><a class="txt" href="#E7:28_1_1_1"><i><font color="Green" title="E19">A10</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">u</font>
<b>by </b><i><a class="txt" href="#E7:28_1_1_1"><i><font color="Green" title="E19">A10</font></i></a>, <a class="txt" href="#E7:28_1_1_1_3"><i><font color="Green" title="E27">A20</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E9:28_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="#E6:28_1_1_1"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="#E9:28_1_1_1"><i><font color="Green" title="E21">A12</font></i></a>, <a class="txt" href="#E1:28_1_1_1_3"><i><font color="Green" title="E23">A14</font></i></a>, <a class="txt" href="#E5:28_1_1_1_3"><i><font color="Green" title="E26">A17</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E12:28_1_1_1"/><b>then </b><i><font color="Green" title="E23">A21</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 <b>by </b><i><a class="txt" href="#E10:28_1_1_1"><i><font color="Green" title="E22">A13</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><a NAME="E13:28_1_1_1"/><b>then </b><i><font color="Green" title="E24">A22</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E14:28_1_1_1"/><i><font color="Green" title="E25">A23</font></i>: 
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> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E12:28_1_1_1"><i><font color="Green" title="E23">A21</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E15:28_1_1_1"/>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <b>by </b><i><a class="txt" href="#E9:28_1_1_1"><i><font color="Green" title="E21">A12</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E16:28_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <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"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E14:28_1_1_1"><i><font color="Green" title="E25">A23</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E17:28_1_1_1"/><b>then </b><i><font color="Green" title="E26">A24</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="#E7:28_1_1_1"><i><font color="Green" title="E19">A10</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E18:28_1_1_1"/><i><font color="Green" title="E27">A25</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <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"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E9:28_1_1_1"><i><font color="Green" title="E21">A12</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E19:28_1_1_1"/><b>then </b><i><font color="Green" title="E28">A26</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="#E7:28_1_1_1"><i><font color="Green" title="E19">A10</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E20:28_1_1_1"/><i><font color="Green" title="E29">A27</font></i>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/><a NAME="E21:28_1_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E22:28_1_1_1"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E9:28_1_1_1"><i><font color="Green" title="E21">A12</font></i></a>, <a class="txt" href="#E19:28_1_1_1"><i><font color="Green" title="E28">A26</font></i></a>, <a class="txt" href="#E20:28_1_1_1"><i><font color="Green" title="E29">A27</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><b>then </b><i><font color="Green" title="E30">A28</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c3">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="#E9:28_1_1_1"><i><font color="Green" title="E21">A12</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c3">m</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c6">b</font>
<b>by </b><i><a class="txt" href="#E7:28_1_1_1"><i><font color="Green" title="E19">A10</font></i></a>, <a class="txt" href="#E18:28_1_1_1"><i><font color="Green" title="E27">A25</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
<br/>.= 
 <a href="polynom7.html#K3" title="POLYNOM7:func.3">coefficient</a> <font color="Maroon" title="c3">m</font>
<b>by </b><i><a class="txt" href="#E1:28_1_1_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="polynom7.html#D6" target="_self" title="POLYNOM7:def.6">Def6</a></i>
;<br/> <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">y</font> = 
 <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="c5">y</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:28"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/><b>then </b><font color="Maroon" title="c5">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<font color="Maroon" title="c5">y</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<b>by </b><i><a class="txt" href="#E14:28_1_1_1"><i><font color="Green" title="E25">A23</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="polynom2.html#K4" title="POLYNOM2:func.4">@</a> </span>)</span>,<font color="Maroon" title="c4">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:28"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="#E13:28_1_1_1"><i><font color="Green" title="E24">A22</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">sm</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c4">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E17:28_1_1_1"><i><font color="Green" title="E26">A24</font></i></a>, <a class="ref" href="polynom2.html#D3" title="POLYNOM2:def.3">POLYNOM2:def 3</a></i>
;<br/><a NAME="E26:28_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K3" title="POLYNOM7:func.3">coefficient</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c4">x</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="#E2:28"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="#E13:28_1_1_1"><i><font color="Green" title="E24">A22</font></i></a>, <a class="txt" href="#E23:28_1_1_1"><i><font color="Green" title="E30">A28</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/><b>hence </b> <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">x</font> = 
<span class="p1">(<span class="default"><a href="polynom7.html#K3" title="POLYNOM7:func.3">coefficient</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c4">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:28"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="rlvect_1.html#T61" title="RLVECT_1:th.61">RLVECT_1:61</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom7.html#K3" title="POLYNOM7:func.3">coefficient</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c4">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:28_1_1_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="polynom7.html#D6" target="_self" title="POLYNOM7:def.6">Def6</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_2"><b>case </b></a><a NAME="E1:28_1_1_2"/><i><font color="Green" title="E15">A29</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K3" title="POLYNOM7:func.3">coefficient</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c4">x</font></span>)</span></span><br/><div class="add"><a NAME="E2:28_1_1_2"/><i><font color="Green" title="E16">A30</font></i>: 
 <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_1_1_2_1"/>
 <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then reconsider </b><font color="Maroon" title="c7">sm</font> =  <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">m</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="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/>
<b>consider </b><font color="Maroon" title="c8">c</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of <font color="Maroon" title="c7">sm</font>;<br/>
<a NAME="E4:28_1_1_2_1"/>
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">m</font>
 
;<br/>
<a NAME="E5:28_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c3">m</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c8">c</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>
 
<b>by </b><i><a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E6:28_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E4:28"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="#E1:28_1_1_2"><i><font color="Green" title="E15">A29</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E3:28_1_1_2"/><b>then </b><i><font color="Green" title="E17">A31</font></i>: 
 <a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">EmptyBag</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="polynom7.html#D6" target="_self" title="POLYNOM7:def.6">Def6</a></i>;<br/><a NAME="E4:28_1_1_2"/>
<font color="Maroon" title="c3">m</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <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>
 <b>by </b><i><a class="txt" href="#E2:28_1_1_2"><i><font color="Green" title="E16">A30</font></i></a>, <a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/><a NAME="E5:28_1_1_2"/><b>then </b><i><font color="Green" title="E18">A32</font></i>: 
<span class="p1">(<span class="default"><a href="polynom7.html#K3" title="POLYNOM7:func.3">coefficient</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c4">x</font></span>)</span> <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>
 <b>by </b><i><a class="txt" href="#E3:28_1_1_2"><i><font color="Green" title="E17">A31</font></i></a>, <a class="ref" href="vectsp_1.html#T39" title="VECTSP_1:th.39">VECTSP_1:39</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">y</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="c2">L</font><b> such that </b><br/><a NAME="E7:28_1_1_2"/><i><font color="Green" title="E19">A33</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">y</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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> &amp;  <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">x</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="c7">y</font> &amp; ( 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 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</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="c7">y</font> holds <br/><font color="Maroon" title="c7">y</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="polynom2.html#K4" title="POLYNOM2:func.4">@</a> </span>)</span>,<font color="Maroon" title="c4">x</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="polynom2.html#D4" title="POLYNOM2:def.4">POLYNOM2:def 4</a></i>;<br/><a NAME="E8:28_1_1_2"/>
 <a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">m</font>
 <b>by </b><i><a class="ref" href="polynom2.html#T20" title="POLYNOM2:th.20">POLYNOM2:20</a></i>;<br/><a NAME="E9:28_1_1_2"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="#E2:28_1_1_2"><i><font color="Green" title="E16">A30</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>;<br/><a NAME="E10:28_1_1_2"/><b>then </b>
 <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></i>;<br/><a NAME="E11:28_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E7:28_1_1_2"><i><font color="Green" title="E19">A33</font></i></a></i>;<br/><a NAME="E12:28_1_1_2"/><b>then </b>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>
 ;<br/><b>hence </b><a NAME="E13:28_1_1_2"/>
 <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K3" title="POLYNOM7:func.3">coefficient</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c4">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:28_1_1_2"><i><font color="Green" title="E18">A32</font></i></a>, <a class="txt" href="#E7:28_1_1_2"><i><font color="Green" title="E19">A33</font></i></a>, <a class="ref" href="rlvect_1.html#T60" title="RLVECT_1:th.60">RLVECT_1:60</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:28"/>
 <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K3" title="POLYNOM7:func.3">coefficient</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c4">x</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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