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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>;<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="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> <a href="group_1.html#V2" title="GROUP_1:attr.2">unital</a> <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a> <a href="group_1.html#V6" title="GROUP_1:attr.6">commutative</a> <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">distributive</a> <a href="algstr_1.html#V1" title="ALGSTR_1:attr.1">left_zeroed</a> <a href="algstr_1.html#V2" title="ALGSTR_1:attr.2">add-left-cancelable</a> <a href="vectsp_2.html#V1" title="VECTSP_2:attr.1">domRing-like</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c3">p</font> be   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>;<br/><b>let </b><font color="Maroon" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font>;<br/><b>let </b><font color="Maroon" title="c5">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>;<br/>









<b>consider </b><font color="Maroon" title="c6">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="E2:61"/><i><font color="Green" title="E28">A1</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">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#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span>,<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c6">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="c6">y</font> holds <br/><font color="Maroon" title="c6">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"><span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span> <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#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span></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="group_1.html#K10" title="GROUP_1:func.10">*</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#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</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="polynom2.html#K4" title="POLYNOM2:func.4">@</a> </span>)</span>,<font color="Maroon" title="c5">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="c7">z</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="E4:61"/><i><font color="Green" title="E29">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">z</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#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> &amp;  <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c7">z</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">z</font> holds <br/><font color="Maroon" title="c7">z</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">p</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#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c3">p</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="group_1.html#K10" title="GROUP_1:func.10">*</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#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c3">p</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="c5">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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_1_1"/>
( <font color="Maroon" title="c4">a</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> or <font color="Maroon" title="c4">a</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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_1"><b>case </b></a><a NAME="E1:61_1_1_1"/><i><font color="Green" title="E30">A3</font></i>: 
<font color="Maroon" title="c4">a</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>
 ;<br/><div class="add"><div><i><font color="Green" title="E31">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">b</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c8">b</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="ref" href="polynom7.html#D11" target="_self" title="POLYNOM7:def.11">Def11</a></i>
<br/>.= 
 <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="polynom7.html#E1:61_1_1_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="binom.html#T2" title="BINOM:th.2">BINOM:2</a></i>
;<br/></div><b>end;</b></div><div><i><font color="Green" title="E32">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_1_1_1_2"/>
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><b>then reconsider </b><font color="Maroon" title="c8">sp</font> =  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span> 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#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/><b>consider </b><font color="Maroon" title="c9">c</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of <font color="Maroon" title="c8">sp</font>;<br/><a NAME="E4:61_1_1_1_2"/>
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">sp</font>
 ;<br/><a NAME="E5:61_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c9">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="E6:61_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom7.html#E2:61_1_1_1"><i><font color="Green" title="E31">A4</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E4:61_1_1_1"/>
 <a href="polynom1.html#K17" title="POLYNOM1:func.17">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#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom2.html#T20" title="POLYNOM2:th.20">POLYNOM2:20</a></i>;<br/><a NAME="E5:61_1_1_1"/><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#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span></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="polynom7.html#E3:61_1_1_1"><i><font color="Green" title="E32">A5</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="E6:61_1_1_1"/><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#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</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="ref" href="finseq_1.html#T27" title="FINSEQ_1:th.27">FINSEQ_1:27</a></i>;<br/><a NAME="E7:61_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">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="polynom7.html#E2:61"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/><a NAME="E8:61_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">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>
 <b>by </b><i><a class="ref" href="finseq_1.html#T32" title="FINSEQ_1:th.32">FINSEQ_1:32</a></i>;<br/><b>then </b> <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c6">y</font> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T60" title="RLVECT_1:th.60">RLVECT_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c7">z</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="txt" href="polynom7.html#E1:61_1_1_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="binom.html#T2" title="BINOM:th.2">BINOM:2</a></i>
;<br/><b>hence </b><a NAME="E10:61_1_1_1"/>
 <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span>,<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="polynom7.html#E2:61"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="polynom7.html#E4:61"><i><font color="Green" title="E29">A2</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_2"><b>case </b></a><a NAME="E1:61_1_1_2"/><i><font color="Green" title="E30">A6</font></i>: 
<font color="Maroon" title="c4">a</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>
 ;<br/><div class="add"><a NAME="E2:61_1_1_2"/><i><font color="Green" title="E31">A7</font></i>: 
 <a href="polynom1.html#K17" title="POLYNOM1:func.17">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#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom2.html#T20" title="POLYNOM2:th.20">POLYNOM2:20</a></i>;<br/><a NAME="E3:61_1_1_2"/><i><font color="Green" title="E32">A8</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#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c3">p</font> holds <br/><font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:61_1_1_2_1"/><i><font color="Green" title="E33">A9</font></i>: 
<font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c3">p</font>
 ;<br/>

<a NAME="E2:61_1_1_2_1"/><b>then </b><i><font color="Green" title="E34">A10</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">u'</font> = <font color="Maroon" title="c8">u</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="txt" href="polynom7.html#E1:61_1_1_2_1"><i><font color="Green" title="E33">A9</font></i></a></i>;<br/>
<a NAME="E4:61_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c9">u'</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c4">a</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="polynom7.html#E1:61_1_1_2"><i><font color="Green" title="E30">A6</font></i></a>, <a class="txt" href="polynom7.html#E2:61_1_1_2_1"><i><font color="Green" title="E34">A10</font></i></a>, <a class="ref" href="vectsp_2.html#D5" title="VECTSP_2:def.5">VECTSP_2:def 5</a></i>;<br/>
<a NAME="E5:61_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c9">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="ref" href="polynom7.html#D11" target="_self" title="POLYNOM7:def.11">Def11</a></i>;<br/>
<b>hence </b><a NAME="E6:61_1_1_2_1"/>
<font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:61_1_1_2"/><i><font color="Green" title="E33">A11</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#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</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#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c3">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:61_1_1_2_2"/><i><font color="Green" title="E34">A12</font></i>: 
<font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c9">u'</font> = <font color="Maroon" title="c8">u</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> ;<br/>
<a NAME="E3:61_1_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="polynom7.html#E1:61_1_1_2_2"><i><font color="Green" title="E34">A12</font></i></a>, <a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/>
<a NAME="E4:61_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c9">u'</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c4">a</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="polynom7.html#D11" target="_self" title="POLYNOM7:def.11">Def11</a></i>;<br/>
<a NAME="E5:61_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c9">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="ref" href="binom.html#T1" title="BINOM:th.1">BINOM:1</a></i>;<br/>
<b>hence </b><a NAME="E6:61_1_1_2_2"/>
<font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:61_1_1_2"/><b>then </b><i><font color="Green" title="E34">A13</font></i>: 
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="txt" href="polynom7.html#E3:61_1_1_2"><i><font color="Green" title="E32">A8</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><a NAME="E6:61_1_1_2"/><i><font color="Green" title="E35">A14</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="txt" href="polynom7.html#E2:61"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="polynom7.html#E4:61"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="polynom7.html#E3:61_1_1_2"><i><font color="Green" title="E32">A8</font></i></a>, <a class="txt" href="polynom7.html#E4:61_1_1_2"><i><font color="Green" title="E33">A11</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><b>then </b><i><font color="Green" title="E36">A15</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">z</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="c6">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> <font color="Maroon" title="c6">y</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:61_1_1_2_4"/><i><font color="Green" title="E37">A16</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">z</font>
 ;<br/><a NAME="E2:61_1_1_2_4"/><b>then </b>
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">z</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 NAME="E3:61_1_1_2_4"/><b>then </b>
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</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">k</font> &amp; <font color="Olive" title="b1">k</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">z</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="c9">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:61_1_1_2_4"/><i><font color="Green" title="E38">A17</font></i>: 
( <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font> &amp; <font color="Maroon" title="c9">k</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">z</font> )
 ;<br/><i><font color="Green" title="E39">A18</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c7">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c4">a</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <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#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p0">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p0">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="polynom2.html#K4" title="POLYNOM2:func.4">@</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="txt" href="polynom7.html#E4:61"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="polynom7.html#E5:61_1_1_2"><i><font color="Green" title="E34">A13</font></i></a>, <a class="txt" href="polynom7.html#E5:61_1_1_2_4"><i><font color="Green" title="E38">A17</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <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#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</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#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="polynom2.html#K4" title="POLYNOM2:func.4">@</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
;<br/><i><font color="Green" title="E40">A19</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">z</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> <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#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom7.html#E2:61"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="polynom7.html#E7:61_1_1_2"><i><font color="Green" title="E36">A15</font></i></a>, <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#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span></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 NAME="E8:61_1_1_2_4"/><i><font color="Green" title="E41">A20</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_2_4_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:61_1_1_2_4_3"/>
<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#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E2:61_1_1_2_4_3"/><b>then </b>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom7.html#E2:61_1_1_2"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>;<br/><b>hence </b><a NAME="E3:61_1_1_2_4_3"/>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="txt" href="polynom7.html#E8:61_1_1_2_4"><i><font color="Green" title="E41">A20</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E10:61_1_1_2_4"/><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#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">q</font> = <font color="Maroon" title="c3">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.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#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span></span>)</span> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#T20" title="FINSEQ_1:th.20">FINSEQ_1:20</a></i>;<br/><a NAME="E12:61_1_1_2_4"/>
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="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">q</font> holds <br/><font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_2_4_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:61_1_1_2_4_4"/>
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">q</font>
 ;<br/>

<a NAME="E2:61_1_1_2_4_4"/><b>then </b><i><font color="Green" title="E42">A21</font></i>: 
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T25" title="FUNCT_1:th.25">FUNCT_1:25</a></i>;<br/>
<a NAME="E3:61_1_1_2_4_4"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<b>hence </b><a NAME="E4:61_1_1_2_4_4"/>
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>
 <b>by </b><i><a class="txt" href="polynom7.html#E2:61_1_1_2_4_4"><i><font color="Green" title="E42">A21</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:61_1_1_2_4"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">q</font> = <font color="Maroon" title="c10">q</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font> <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E15:61_1_1_2_4"/>
<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#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</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#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="txt" href="polynom7.html#E1:61_1_1_2_4"><i><font color="Green" title="E37">A16</font></i></a>, <a class="txt" href="polynom7.html#E5:61_1_1_2_4"><i><font color="Green" title="E38">A17</font></i></a>, <a class="txt" href="polynom7.html#E7:61_1_1_2_4"><i><font color="Green" title="E40">A19</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E16:61_1_1_2_4"/><b>then </b>
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">q</font>
 <b>by </b><i><a class="txt" href="polynom7.html#E1:61_1_1_2_4"><i><font color="Green" title="E37">A16</font></i></a>, <a class="txt" href="polynom7.html#E5:61_1_1_2_4"><i><font color="Green" title="E38">A17</font></i></a>, <a class="txt" href="polynom7.html#E7:61_1_1_2_4"><i><font color="Green" title="E40">A19</font></i></a>, <a class="txt" href="polynom7.html#E8:61_1_1_2_4"><i><font color="Green" title="E41">A20</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="E42">A22</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</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#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font> = 
<font color="Maroon" title="c11">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
<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">p</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#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom7.html#E1:61_1_1_2_4"><i><font color="Green" title="E37">A16</font></i></a>, <a class="txt" href="polynom7.html#E5:61_1_1_2_4"><i><font color="Green" title="E38">A17</font></i></a>, <a class="txt" href="polynom7.html#E7:61_1_1_2_4"><i><font color="Green" title="E40">A19</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">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom7.html#E1:61_1_1_2_4"><i><font color="Green" title="E37">A16</font></i></a>, <a class="txt" href="polynom7.html#E5:61_1_1_2_4"><i><font color="Green" title="E38">A17</font></i></a>, <a class="txt" href="polynom7.html#E7:61_1_1_2_4"><i><font color="Green" title="E40">A19</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/><b>reconsider </b><font color="Maroon" title="c12">c</font> = <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#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> ;<br/><b>reconsider </b><font color="Maroon" title="c13">c</font> = <font color="Maroon" title="c12">c</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> ;<br/><i><font color="Green" title="E43">A23</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c13">c</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</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#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="txt" href="polynom7.html#E17:61_1_1_2_4"><i><font color="Green" title="E42">A22</font></i></a>, <a class="ref" href="polynom7.html#D11" target="_self" title="POLYNOM7:def.11">Def11</a></i>
;<br/><a NAME="E21:61_1_1_2_4"/><i><font color="Green" title="E44">A24</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_2_4_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:61_1_1_2_4_7"/>
<font color="Maroon" title="c14">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#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E2:61_1_1_2_4_7"/><b>then </b>
<font color="Maroon" title="c14">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom7.html#E2:61_1_1_2"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>;<br/><b>hence </b><a NAME="E3:61_1_1_2_4_7"/>
<font color="Maroon" title="c14">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom7.html#E21:61_1_1_2_4"><i><font color="Green" title="E44">A24</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E23:61_1_1_2_4"/><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#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c14">r</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.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#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span></span>)</span> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#T20" title="FINSEQ_1:th.20">FINSEQ_1:20</a></i>;<br/><a NAME="E25:61_1_1_2_4"/>
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="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">r</font> holds <br/><font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_2_4_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:61_1_1_2_4_8"/>
<font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">r</font>
 ;<br/>

<a NAME="E2:61_1_1_2_4_8"/><b>then </b><i><font color="Green" title="E45">A25</font></i>: 
<font color="Maroon" title="c15">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"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T25" title="FUNCT_1:th.25">FUNCT_1:25</a></i>;<br/>
<a NAME="E3:61_1_1_2_4_8"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<b>hence </b><a NAME="E4:61_1_1_2_4_8"/>
<font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>
 <b>by </b><i><a class="txt" href="polynom7.html#E2:61_1_1_2_4_8"><i><font color="Green" title="E45">A25</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E26:61_1_1_2_4"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c15">r</font> = <font color="Maroon" title="c14">r</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font> <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E28:61_1_1_2_4"/>
<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#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</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#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="txt" href="polynom7.html#E1:61_1_1_2_4"><i><font color="Green" title="E37">A16</font></i></a>, <a class="txt" href="polynom7.html#E5:61_1_1_2_4"><i><font color="Green" title="E38">A17</font></i></a>, <a class="txt" href="polynom7.html#E7:61_1_1_2_4"><i><font color="Green" title="E40">A19</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E29:61_1_1_2_4"/><b>then </b>
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">r</font>
 <b>by </b><i><a class="txt" href="polynom7.html#E1:61_1_1_2_4"><i><font color="Green" title="E37">A16</font></i></a>, <a class="txt" href="polynom7.html#E5:61_1_1_2_4"><i><font color="Green" title="E38">A17</font></i></a>, <a class="txt" href="polynom7.html#E7:61_1_1_2_4"><i><font color="Green" title="E40">A19</font></i></a>, <a class="txt" href="polynom7.html#E21:61_1_1_2_4"><i><font color="Green" title="E44">A24</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><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span> <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#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font> = 
<font color="Maroon" title="c15">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span> <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#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom7.html#E1:61_1_1_2_4"><i><font color="Green" title="E37">A16</font></i></a>, <a class="txt" href="polynom7.html#E5:61_1_1_2_4"><i><font color="Green" title="E38">A17</font></i></a>, <a class="txt" href="polynom7.html#E7:61_1_1_2_4"><i><font color="Green" title="E40">A19</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</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#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="txt" href="polynom7.html#E1:61_1_1_2_4"><i><font color="Green" title="E37">A16</font></i></a>, <a class="txt" href="polynom7.html#E5:61_1_1_2_4"><i><font color="Green" title="E38">A17</font></i></a>, <a class="txt" href="polynom7.html#E7:61_1_1_2_4"><i><font color="Green" title="E40">A19</font></i></a>, <a class="txt" href="polynom7.html#E20:61_1_1_2_4"><i><font color="Green" title="E43">A23</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/><b>hence </b><a NAME="E31:61_1_1_2_4"/>
<font color="Maroon" title="c6">y</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="polynom7.html#E2:61"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="polynom7.html#E6:61_1_1_2"><i><font color="Green" title="E35">A14</font></i></a>, <a class="txt" href="polynom7.html#E5:61_1_1_2_4"><i><font color="Green" title="E38">A17</font></i></a>, <a class="txt" href="polynom7.html#E6:61_1_1_2_4"><i><font color="Green" title="E39">A18</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E9:61_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">*</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="polynom7.html#E7:61_1_1_2"><i><font color="Green" title="E36">A15</font></i></a>, <a class="ref" href="polynom1.html#D3" title="POLYNOM1:def.3">POLYNOM1:def 3</a></i>;<br/><b>hence </b><a NAME="E10:61_1_1_2"/>
 <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span>,<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="polynom7.html#E2:61"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="polynom7.html#E4:61"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="binom.html#T5" title="BINOM:th.5">BINOM:5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:61"/>
 <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom7.html#K6" title="POLYNOM7:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span>,<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c4">a</font>
 ;<br/>


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