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<b>let </b><font color="Maroon">n</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V2">unital</a> <a href="vectsp_1.html#V7">distributive</a> non <a href="realset2.html#V3">trivial</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>, <font color="Maroon">q</font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM5">Function</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>;<br/>









<b>assume </b><a NAME="E1:48"/><i><font color="Green">E32</font></i>: 
( not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font> &amp;  <a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">b</font></span><a href="tarski.html#K1">}</a></span> &amp; ( for <font color="Olive">b'</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>  st <font color="Olive">b'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> holds <br/><font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b'</font> ) )
 ;<br/>

<b>set </b><font color="Maroon">sq</font> =  <a href="triang_1.html#K2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">sp</font> =  <a href="triang_1.html#K2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span>;<br/>
<a NAME="E2:48"/><i><font color="Green">E34</font></i>: 
(  <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font> <a href="orders_1.html#R3">linearly_orders</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font> &amp;  <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font> <a href="orders_1.html#R3">linearly_orders</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:48"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">b</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:48"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="txt" href="polynom2.html#E7:48"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:48"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom2.html#E8:48"><i><font color="Green">E36</font></i></a>, <a class="ref" href="triang_1.html#D2">TRIANG_1:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:48"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E8:48"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="txt" href="polynom2.html#E10:48"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E9:48"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:48"/><b>then </b><i><font color="Green">E37</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="polynom2.html#E10:48"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E11:48"/><b>then </b>
1 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">k'</font> = <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/>
<a NAME="E14:48"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">k'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 0
 
;<br/>
<a NAME="E15:48"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="triang_1.html#K2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_5.html#K5">Ins</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span>,<font color="Maroon">k'</font>,<font color="Maroon">b</font>
 
<b>by </b><i><a class="txt" href="polynom2.html#E7:48"><i><font color="Green">E35</font></i></a>, <a class="txt" href="polynom2.html#E8:48"><i><font color="Green">E36</font></i></a>, <a class="txt" href="polynom2.html#E10:48"><i><font color="Green">E37</font></i></a>, , </i>;<br/>
<b>consider </b><font color="Maroon">yq</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E17:48"/><i><font color="Green">E45</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">yq</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font></span>)</span></span>)</span> &amp;  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">yq</font> <a href="hidden.html#R1">=</a>  <a href="polynom2.html#K5" target="_self">eval</a> <font color="Maroon">q</font>,<font color="Maroon">x</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">yq</font> holds <br/><font color="Maroon">yq</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" target="_self">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">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="polynom2.html#K4" target="_self">@</a> </span>)</span>,<font color="Maroon">x</font></span>)</span> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">yp</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E19:48"/><i><font color="Green">E46</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">yp</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> &amp;  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">yp</font> <a href="hidden.html#R1">=</a>  <a href="polynom2.html#K5" target="_self">eval</a> <font color="Maroon">p</font>,<font color="Maroon">x</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">yp</font> holds <br/><font color="Maroon">yp</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" target="_self">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">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="polynom2.html#K4" target="_self">@</a> </span>)</span>,<font color="Maroon">x</font></span>)</span> ) )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> ;<br/>
<b>set </b><font color="Maroon">ytmp</font> =  <a href="finseq_5.html#K5">Ins</a> <font color="Maroon">yp</font>,<font color="Maroon">k'</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><a href="polynom2.html#K3" target="_self">eval</a> <font color="Maroon">b</font>,<font color="Maroon">x</font></span>)</span></span>)</span>;<br/>
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">yp</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k'</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p5">(<span class="default"><a href="polynom2.html#K3" target="_self">eval</a> <font color="Maroon">b</font>,<font color="Maroon">x</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">yp</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k'</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">yp</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k'</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p5">(<span class="default"><a href="polynom2.html#K3" target="_self">eval</a> <font color="Maroon">b</font>,<font color="Maroon">x</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">yp</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">yp</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k'</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p5">(<span class="default"><a href="polynom2.html#K3" target="_self">eval</a> <font color="Maroon">b</font>,<font color="Maroon">x</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">yp</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">yp</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k'</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">yp</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k'</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><a href="polynom2.html#K3" target="_self">eval</a> <font color="Maroon">b</font>,<font color="Maroon">x</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">yp</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k'</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><font color="Maroon">yp</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k'</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><a href="polynom2.html#K3" target="_self">eval</a> <font color="Maroon">b</font>,<font color="Maroon">x</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">yp</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><a href="polynom2.html#K3" target="_self">eval</a> <font color="Maroon">b</font>,<font color="Maroon">x</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rfinseq.html#T21">RFINSEQ:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom2.html#K5" target="_self">eval</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><a href="polynom2.html#K3" target="_self">eval</a> <font color="Maroon">b</font>,<font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T61">RLVECT_1:61</a></i>
;<br/>
<a NAME="E22:48"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">yp</font>,<font color="Maroon">k'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" target="_self">eval</a> <font color="Maroon">b</font>,<font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K5" target="_self">eval</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><a href="polynom2.html#K3" target="_self">eval</a> <font color="Maroon">b</font>,<font color="Maroon">x</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_5.html#D4">FINSEQ_5:def 4</a></i>;<br/>
<i><font color="Green">E57</font></i>: <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="polynom2.html#E8:48"><i><font color="Green">E36</font></i></a>, <a class="ref" href="triang_1.html#T9">TRIANG_1:9</a></i>
<br/>.= 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E7:48"><i><font color="Green">E35</font></i></a>, <a class="ref" href="card_2.html#T54">CARD_2:54</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E8:48"><i><font color="Green">E36</font></i></a>, <a class="ref" href="triang_1.html#T9">TRIANG_1:9</a></i>
;<br/>
<a NAME="E24:48"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">yq</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">yp</font>,<font color="Maroon">k'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" target="_self">eval</a> <font color="Maroon">b</font>,<font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom2.html#E15:48"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="finseq_5.html#T72">FINSEQ_5:72</a></i>;<br/>
<a NAME="E25:48"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">yq</font> holds <br/><font color="Maroon">yq</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">yp</font>,<font color="Maroon">k'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" target="_self">eval</a> <font color="Maroon">b</font>,<font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:48_3"/><i><font color="Green">E59</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">yq</font> )
 ;<br/>

<a NAME="E2:48_3"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">yq</font></span>)</span> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">yp</font>,<font color="Maroon">k'</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><a href="polynom2.html#K3" target="_self">eval</a> <font color="Maroon">b</font>,<font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="polynom2.html#E19:48"><i><font color="Green">E46</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:48_3"/><b>then </b><i><font color="Green">E60</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">yq</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">yp</font>,<font color="Maroon">k'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" target="_self">eval</a> <font color="Maroon">b</font>,<font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_3_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_3_1_1"><b>suppose </b></a><a NAME="E1:48_3_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:48_3_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:48_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="txt" href="polynom2.html#E10:48"><i><font color="Green">E37</font></i></a>, <a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><a NAME="E4:48_3_1_1"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="polynom2.html#E10:48"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><b>then </b><i><font color="Green">E63</font></i>: <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font>
<b>by </b><i><a class="txt" href="polynom2.html#E10:48"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
;<br/><i><font color="Green">E64</font></i>: <font color="Maroon">yq</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" target="_self">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">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> <a href="polynom2.html#K4" target="_self">@</a> </span>)</span>,<font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E15:48"><i><font color="Green">E42</font></i></a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" target="_self">eval</a> <font color="Maroon">b</font>,<font color="Maroon">x</font></span>)</span>
<b>by </b><i>, </i>
;<br/><a NAME="E7:48_3_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">k'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">yp</font>
 <b>by </b><i><a class="txt" href="polynom2.html#E14:48"><i><font color="Green">E39</font></i></a>, , , <a class="txt" href="polynom2.html#E17:48"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">yp</font>,<font color="Maroon">k'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" target="_self">eval</a> <font color="Maroon">b</font>,<font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">yp</font>,<font color="Maroon">k'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" target="_self">eval</a> <font color="Maroon">b</font>,<font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" target="_self">eval</a> <font color="Maroon">b</font>,<font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="finseq_5.html#T76">FINSEQ_5:76</a></i>
<br/>.= 
<font color="Maroon">yq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_3_1_2"><b>suppose </b></a><a NAME="E1:48_3_1_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:48_3_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span>,<font color="Maroon">k'</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_5.html#T72">FINSEQ_5:72</a></i>;<br/><a NAME="E3:48_3_1_2"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span>,<font color="Maroon">k'</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_3_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_3_1_2_1_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> or <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_3_1_2_1_1_1"><b>case </b></a><a NAME="E1:48_3_1_2_1_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:48_3_1_2_1_1_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k'</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:48_3_1_2_1_1_1"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k'</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:48_3_1_2_1_1_1"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="polynom2.html#E14:48"><i><font color="Green">E39</font></i></a>, <a class="txt" href="polynom2.html#E17:48"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E5:48_3_1_2_1_1_1"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:48_3_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k'</font></span>)</span> is   <a href="finseq_1.html#NM1">FinSequence</a>
 <b>by </b><i><a class="ref" href="finseq_1.html#T19">FINSEQ_1:19</a></i>;<br/><a NAME="E7:48_3_1_2_1_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k'</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/><a NAME="E8:48_3_1_2_1_1_1"/><b>then </b><i><font color="Green">E109</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k'</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/><a NAME="E9:48_3_1_2_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E10:48_3_1_2_1_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E11:48_3_1_2_1_1_1"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span>,<font color="Maroon">k'</font>,<font color="Maroon">b</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E12:48_3_1_2_1_1_1"/><b>then </b><i><font color="Green">E112</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span>,<font color="Maroon">k'</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span>,<font color="Maroon">k'</font>,<font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E13:48_3_1_2_1_1_1"/><i><font color="Green">E113</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span>,<font color="Maroon">k'</font>,<font color="Maroon">b</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E14:48_3_1_2_1_1_1"/><i><font color="Green">E114</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> is   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><a NAME="E15:48_3_1_2_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E16:48_3_1_2_1_1_1"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E17:48_3_1_2_1_1_1"/><b>then </b><i><font color="Green">E125</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E18:48_3_1_2_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E19:48_3_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E20:48_3_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E21:48_3_1_2_1_1_1"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><div><i><font color="Green">E127</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_3_1_2_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E2:48_3_1_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E3:48_3_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:48_3_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom2.html#E7:48"><i><font color="Green">E35</font></i></a>, <a class="txt" href="polynom2.html#E8:48"><i><font color="Green">E36</font></i></a>, <a class="ref" href="triang_1.html#D2">TRIANG_1:def 2</a></i>;<br/></div><b>end;</b></div><a NAME="E23:48_3_1_2_1_1_1"/>
<font color="Maroon">k'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T59">NAT_1:59</a></i>;<br/><a NAME="E24:48_3_1_2_1_1_1"/><b>then </b>
<font color="Maroon">k'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">yq</font>
 <b>by </b><i><a class="txt" href="polynom2.html#E14:48"><i><font color="Green">E39</font></i></a>, <a class="txt" href="polynom2.html#E15:48"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E25:48_3_1_2_1_1_1"/><b>then </b><i><font color="Green">E128</font></i>: 
<font color="Maroon">k'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">yp</font>
 <b>by </b><i><a class="txt" href="polynom2.html#E15:48"><i><font color="Green">E42</font></i></a>, , <a class="txt" href="polynom2.html#E17:48"><i><font color="Green">E45</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E26:48_3_1_2_1_1_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">yq</font>
 <b>by </b><i><a class="txt" href="polynom2.html#E14:48"><i><font color="Green">E39</font></i></a>, <a class="txt" href="polynom2.html#E15:48"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E27:48_3_1_2_1_1_1"/><b>then </b><i><font color="Green">E129</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">yp</font>
 <b>by </b><i><a class="txt" href="polynom2.html#E15:48"><i><font color="Green">E42</font></i></a>, , <a class="txt" href="polynom2.html#E17:48"><i><font color="Green">E45</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E28:48_3_1_2_1_1_1"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k'</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E29:48_3_1_2_1_1_1"/>
<font color="Maroon">yp</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k'</font></span>)</span> is   <a href="finseq_1.html#NM1">FinSequence</a>
 <b>by </b><i><a class="ref" href="finseq_1.html#T19">FINSEQ_1:19</a></i>;<br/><a NAME="E30:48_3_1_2_1_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">yp</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k'</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="polynom2.html#E3:48_3_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/><a NAME="E31:48_3_1_2_1_1_1"/><b>then </b><i><font color="Green">E131</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">yp</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k'</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/><a NAME="E32:48_3_1_2_1_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E33:48_3_1_2_1_1_1"/><b>then </b><i><font color="Green">E132</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span>,<font color="Maroon">k'</font>,<font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="polynom2.html#E23:48"><i><font color="Green">E57</font></i></a>, <a class="txt" href="polynom2.html#E24:48"><i><font color="Green">E58</font></i></a>, <a class="txt" href="polynom2.html#E1:48_3"><i><font color="Green">E59</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><b>then </b><i><font color="Green">E133</font></i>: <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span>,<font color="Maroon">k'</font>,<font color="Maroon">b</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span>,<font color="Maroon">k'</font>,<font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span>,<font color="Maroon">k'</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E2:48_3_1_2_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span>,<font color="Maroon">k'</font>,<font color="Maroon">b</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E23:48"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E22:48"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_5.html#T75">FINSEQ_5:75</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E7:48"><i><font color="Green">E35</font></i></a>, <a class="txt" href="polynom2.html#E3:48_3"><i><font color="Green">E60</font></i></a>, </i>
<br/>.= 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><i><font color="Green">E134</font></i>: <font color="Maroon">yq</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" target="_self">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">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="polynom2.html#K4" target="_self">@</a> </span>)</span>,<font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E15:48"><i><font color="Green">E42</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" target="_self">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">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="polynom2.html#K4" target="_self">@</a> </span>)</span>,<font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="txt" href="polynom2.html#E22:48"><i><font color="Green">E56</font></i></a>, <a class="txt" href="polynom2.html#E3:48_3_1_2_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_5.html#T75">FINSEQ_5:75</a></i>
<br/>.= 
<font color="Maroon">yp</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i>, , <a class="txt" href="polynom2.html#E1:48_3_1_2"><i><font color="Green">E66</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">yp</font>,<font color="Maroon">k'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" target="_self">eval</a> <font color="Maroon">b</font>,<font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="txt" href="polynom2.html#E1:48_3_1_2_1_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_5.html#T75">FINSEQ_5:75</a></i>
;<br/><b>thus </b><font color="Maroon">yq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">yq</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">yp</font>,<font color="Maroon">k'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" target="_self">eval</a> <font color="Maroon">b</font>,<font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_3_1_2_1_1_2"><b>case </b></a><a NAME="E1:48_3_1_2_1_1_2"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:48_3_1_2_1_1_2"/><b>then </b><i><font color="Green">E136</font></i>: 
<font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k'</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E3:48_3_1_2_1_1_2"/>
<font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">i1</font> = <font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E5:48_3_1_2_1_1_2"/><i><font color="Green">E137</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 0
 ;<br/><a NAME="E6:48_3_1_2_1_1_2"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">yp</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="polynom2.html#E15:48"><i><font color="Green">E42</font></i></a>, , <a class="txt" href="polynom2.html#E17:48"><i><font color="Green">E45</font></i></a>, , <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E7:48_3_1_2_1_1_2"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k'</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E8:48_3_1_2_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E9:48_3_1_2_1_1_2"/><b>then </b><i><font color="Green">E139</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:48_3_1_2_1_1_2"/><b>then </b>
<font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E11:48_3_1_2_1_1_2"/><b>then </b><i><font color="Green">E140</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E12:48_3_1_2_1_1_2"/><i><font color="Green">E141</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> is   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><div><i><font color="Green">E142</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_3_1_2_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E2:48_3_1_2_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="polynom2.html#E22:48"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E3:48_3_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="polynom2.html#E22:48"><i><font color="Green">E56</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:48_3_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom2.html#E7:48"><i><font color="Green">E35</font></i></a>, <a class="txt" href="polynom2.html#E8:48"><i><font color="Green">E36</font></i></a>, <a class="ref" href="triang_1.html#D2">TRIANG_1:def 2</a></i>;<br/></div><b>end;</b></div><a NAME="E14:48_3_1_2_1_1_2"/><i><font color="Green">E143</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="polynom2.html#E22:48"><i><font color="Green">E56</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E15:48_3_1_2_1_1_2"/><i><font color="Green">E145</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E16:48_3_1_2_1_1_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E17:48_3_1_2_1_1_2"/><b>then </b><i><font color="Green">E146</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="polynom2.html#E22:48"><i><font color="Green">E56</font></i></a>, <a class="txt" href="polynom2.html#E1:48_3"><i><font color="Green">E59</font></i></a>, <a class="txt" href="polynom2.html#E3:48_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E18:48_3_1_2_1_1_2"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span>,<font color="Maroon">k'</font>,<font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom2.html#E15:48"><i><font color="Green">E42</font></i></a>, <a class="txt" href="polynom2.html#E17:48"><i><font color="Green">E45</font></i></a>, , , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E19:48_3_1_2_1_1_2"/><b>then </b><i><font color="Green">E148</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span>,<font color="Maroon">k'</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span>,<font color="Maroon">k'</font>,<font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E20:48_3_1_2_1_1_2"/><i><font color="Green">E149</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span>,<font color="Maroon">k'</font>,<font color="Maroon">b</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E21:48_3_1_2_1_1_2"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E22:48_3_1_2_1_1_2"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">k'</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font>
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E23:48_3_1_2_1_1_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E24:48_3_1_2_1_1_2"/><b>then </b><i><font color="Green">E152</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span>,<font color="Maroon">k'</font>,<font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><b>then </b><i><font color="Green">E153</font></i>: <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span>,<font color="Maroon">k'</font>,<font color="Maroon">b</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span>,<font color="Maroon">k'</font>,<font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span>,<font color="Maroon">k'</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E3:48_3_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span>,<font color="Maroon">k'</font>,<font color="Maroon">b</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span>
<b>by </b><i>, , , <a class="txt" href="polynom2.html#E1:48_3_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_5.html#T77">FINSEQ_5:77</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E7:48"><i><font color="Green">E35</font></i></a>, <a class="txt" href="polynom2.html#E23:48"><i><font color="Green">E57</font></i></a>, <a class="txt" href="polynom2.html#E24:48"><i><font color="Green">E58</font></i></a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E22:48"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><i><font color="Green">E154</font></i>: <font color="Maroon">yq</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" target="_self">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">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="polynom2.html#K4" target="_self">@</a> </span>)</span>,<font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E15:48"><i><font color="Green">E42</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" target="_self">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">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="polynom2.html#K4" target="_self">@</a> </span>)</span>,<font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , , , <a class="txt" href="polynom2.html#E1:48_3_1_2"><i><font color="Green">E66</font></i></a>, , <a class="ref" href="finseq_5.html#T77">FINSEQ_5:77</a></i>
<br/>.= 
<font color="Maroon">yp</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">yp</font>,<font color="Maroon">k'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" target="_self">eval</a> <font color="Maroon">b</font>,<font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i>, , <a class="txt" href="polynom2.html#E1:48_3_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_5.html#T77">FINSEQ_5:77</a></i>
;<br/><b>thus </b><font color="Maroon">yq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">yq</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">yp</font>,<font color="Maroon">k'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" target="_self">eval</a> <font color="Maroon">b</font>,<font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:48_3_1_2"/>
<font color="Maroon">yq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">yp</font>,<font color="Maroon">k'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" target="_self">eval</a> <font color="Maroon">b</font>,<font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E26:48"/>
 <a href="polynom2.html#K5" target="_self">eval</a> <font color="Maroon">q</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K5" target="_self">eval</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><a href="polynom2.html#K3" target="_self">eval</a> <font color="Maroon">b</font>,<font color="Maroon">x</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="polynom2.html#E15:48"><i><font color="Green">E42</font></i></a>, , <a class="txt" href="polynom2.html#E19:48"><i><font color="Green">E46</font></i></a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


</div>
