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<b>let </b><font color="Maroon" title="c1">L</font> be   <a href="vectsp_2.html#NM1" title="VECTSP_2:NM.1">comRing</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">b1</font>, <font color="Olive" title="b2">b2</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> holds   <a href="uproots.html#K10" title="UPROOTS:func.10">poly_with_roots</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Olive" title="b2">b2</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K10" title="UPROOTS:func.10">poly_with_roots</a> <font color="Olive" title="b1">b1</font></span>)</span> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <span class="p1">(<span class="default"><a href="uproots.html#K10" title="UPROOTS:func.10">poly_with_roots</a> <font color="Olive" title="b2">b2</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">b1</font>, <font color="Maroon" title="c3">b2</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="uproots.html#K10" title="UPROOTS:func.10">poly_with_roots</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c3">b2</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K10" title="UPROOTS:func.10">poly_with_roots</a> <font color="Maroon" title="c2">b1</font></span>)</span> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <span class="p1">(<span class="default"><a href="uproots.html#K10" title="UPROOTS:func.10">poly_with_roots</a> <font color="Maroon" title="c3">b2</font></span>)</span></span><br/>



<b>set </b><font color="Maroon" title="c4">b</font> = <font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c3">b2</font>;<br/>
<b>consider </b><font color="Maroon" title="c5">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> , <font color="Maroon" title="c6">s</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E2:90"/><i><font color="Green" title="E71">A1</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c3">b2</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E3:90"/><i><font color="Green" title="E72">A2</font></i>: 
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p1">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c3">b2</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:90"/><i><font color="Green" title="E73">A3</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">f</font> holds <br/><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K9" title="UPROOTS:func.9">fpoly_mult_root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c3">b2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E5:90"/><i><font color="Green" title="E74">A4</font></i>: 
 <a href="uproots.html#K10" title="UPROOTS:func.10">poly_with_roots</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c3">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="uproots.html#D11" target="_self" title="UPROOTS:def.11">Def11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">f1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> , <font color="Maroon" title="c8">s1</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E7:90"/><i><font color="Green" title="E75">A5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c2">b1</font></span>)</span>
 <b>and </b><br/><a NAME="E8:90"/><i><font color="Green" title="E76">A6</font></i>: 
<font color="Maroon" title="c8">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p1">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c2">b1</font></span>)</span>
 <b>and </b><br/><a NAME="E9:90"/><i><font color="Green" title="E77">A7</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">f1</font> holds <br/><font color="Maroon" title="c7">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K9" title="UPROOTS:func.9">fpoly_mult_root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E10:90"/><i><font color="Green" title="E78">A8</font></i>: 
 <a href="uproots.html#K10" title="UPROOTS:func.10">poly_with_roots</a> <font color="Maroon" title="c2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c7">f1</font></span>)</span>
 <b>by </b><i><a class="ref" href="uproots.html#D11" target="_self" title="UPROOTS:def.11">Def11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">f2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> , <font color="Maroon" title="c10">s2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E12:90"/><i><font color="Green" title="E79">A9</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c3">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E13:90"/><i><font color="Green" title="E80">A10</font></i>: 
<font color="Maroon" title="c10">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p1">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c3">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E14:90"/><i><font color="Green" title="E81">A11</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">f2</font> holds <br/><font color="Maroon" title="c9">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K9" title="UPROOTS:func.9">fpoly_mult_root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">s2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">s2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E15:90"/><i><font color="Green" title="E82">A12</font></i>: 
 <a href="uproots.html#K10" title="UPROOTS:func.10">poly_with_roots</a> <font color="Maroon" title="c3">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c9">f2</font></span>)</span>
 <b>by </b><i><a class="ref" href="uproots.html#D11" target="_self" title="UPROOTS:def.11">Def11</a></i>;<br/>
<b>set </b><font color="Maroon" title="c11">ff</font> =  <a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">f</font>;<br/>
<b>set </b><font color="Maroon" title="c12">ff1</font> =  <a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c7">f1</font>;<br/>
<b>set </b><font color="Maroon" title="c13">ff2</font> =  <a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c9">f2</font>;<br/>
<b>set </b><font color="Maroon" title="c14">g</font> = <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c9">f2</font></span>)</span>;<br/>
<a NAME="E16:90"/><i><font color="Green" title="E83">A13</font></i>: 
 <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c3">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c2">b1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c3">b2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#T42" title="POLYNOM1:th.42">POLYNOM1:42</a></i>;<br/>
<a NAME="E17:90"/><i><font color="Green" title="E84">A14</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K4" title="UPROOTS:func.4">degree</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c3">b2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:90"><i><font color="Green" title="E71">A1</font></i></a>, <a class="txt" href="#E3:90"><i><font color="Green" title="E72">A2</font></i></a>, <a class="txt" href="#E4:90"><i><font color="Green" title="E73">A3</font></i></a>, <a class="ref" href="uproots.html#T64" target="_self" title="UPROOTS:th.64">Th64</a></i>;<br/>
<a NAME="E18:90"/><i><font color="Green" title="E85">A15</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c9">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K4" title="UPROOTS:func.4">degree</a> <font color="Maroon" title="c3">b2</font>
 
<b>by </b><i><a class="txt" href="#E12:90"><i><font color="Green" title="E79">A9</font></i></a>, <a class="txt" href="#E13:90"><i><font color="Green" title="E80">A10</font></i></a>, <a class="txt" href="#E14:90"><i><font color="Green" title="E81">A11</font></i></a>, <a class="ref" href="uproots.html#T64" target="_self" title="UPROOTS:th.64">Th64</a></i>;<br/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c9">f2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c7">f1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c9">f2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="uproots.html#K4" title="UPROOTS:func.4">degree</a> <font color="Maroon" title="c2">b1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K4" title="UPROOTS:func.4">degree</a> <font color="Maroon" title="c3">b2</font></span>)</span>
<b>by </b><i><a class="txt" href="#E7:90"><i><font color="Green" title="E75">A5</font></i></a>, <a class="txt" href="#E8:90"><i><font color="Green" title="E76">A6</font></i></a>, <a class="txt" href="#E9:90"><i><font color="Green" title="E77">A7</font></i></a>, <a class="txt" href="#E18:90"><i><font color="Green" title="E85">A15</font></i></a>, <a class="ref" href="uproots.html#T64" target="_self" title="UPROOTS:th.64">Th64</a></i>
<br/>.= 
 <a href="uproots.html#K4" title="UPROOTS:func.4">degree</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c3">b2</font></span>)</span>
<b>by </b><i><a class="ref" href="uproots.html#T17" target="_self" title="UPROOTS:th.17">Th17</a></i>
;<br/>
<a NAME="E20:90"/><b>then </b><i><font color="Green" title="E86">A16</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c9">f2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E17:90"><i><font color="Green" title="E84">A14</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E21:90"/><i><font color="Green" title="E87">A17</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c3">b2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:90"><i><font color="Green" title="E72">A2</font></i></a>, <a class="ref" href="uproots.html#T5" target="_self" title="UPROOTS:th.5">Def1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c9">f2</font></span>)</span></span>)</span>,<font color="Olive">b<sub>1</sub></font></span>)</span></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:90_2_1"/>
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">f</font></span>)</span> or  not <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">f</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_2_1_1"><b>suppose </b></a><a NAME="E1:90_2_1_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c9">f2</font></span>)</span></span>)</span>,<font color="Olive">b<sub>1</sub></font></span>)</span></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c16">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:90_2_1_1"/><i><font color="Green" title="E88">A18</font></i>: 
<font color="Maroon" title="c16">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">f</font></span>)</span>
 <b>and </b><br/><a NAME="E4:90_2_1_1"/><i><font color="Green" title="E89">A19</font></i>: 
<span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/><b>consider </b><font color="Maroon" title="c17">i</font>, <font color="Maroon" title="c18">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:90_2_1_1"/><i><font color="Green" title="E90">A20</font></i>: 
<font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">f</font>
 <b>and </b><br/><a NAME="E7:90_2_1_1"/><i><font color="Green" title="E91">A21</font></i>: 
<font color="Maroon" title="c18">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font></span>)</span>
 <b>and </b><br/><a NAME="E8:90_2_1_1"/>
<font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K4" title="POLYNOM1:func.4">Card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">j</font>
 <b>and </b><br/><a NAME="E9:90_2_1_1"/><i><font color="Green" title="E92">A22</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font>
 <b>by </b><i><a class="txt" href="#E3:90_2_1_1"><i><font color="Green" title="E88">A18</font></i></a>, <a class="ref" href="polynom1.html#T32" title="POLYNOM1:th.32">POLYNOM1:32</a></i>;<br/><a NAME="E10:90_2_1_1"/>
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K9" title="UPROOTS:func.9">fpoly_mult_root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c3">b2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:90"><i><font color="Green" title="E73">A3</font></i></a>, <a class="txt" href="#E6:90_2_1_1"><i><font color="Green" title="E90">A20</font></i></a></i>;<br/><a NAME="E11:90_2_1_1"/><b>then </b><i><font color="Green" title="E93">A23</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span>
 <b>by </b><i><a class="txt" href="#E7:90_2_1_1"><i><font color="Green" title="E91">A21</font></i></a>, <a class="ref" href="uproots.html#D10" target="_self" title="UPROOTS:def.10">Def10</a></i>;<br/><a NAME="E12:90_2_1_1"/>
<font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">s</font>
 <b>by </b><i><a class="txt" href="#E2:90"><i><font color="Green" title="E71">A1</font></i></a>, <a class="txt" href="#E21:90"><i><font color="Green" title="E87">A17</font></i></a>, <a class="txt" href="#E6:90_2_1_1"><i><font color="Green" title="E90">A20</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E13:90_2_1_1"/><b>then </b>
( <font color="Maroon" title="c6">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font> &amp; <font color="Maroon" title="c6">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">s</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E14:90_2_1_1"/><b>then </b><i><font color="Green" title="E94">A24</font></i>: 
<font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c3">b2</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:90"><i><font color="Green" title="E72">A2</font></i></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/><a NAME="E15:90_2_1_1"/><i><font color="Green" title="E95">A25</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c9">f2</font></span>)</span></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c9">f2</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T63" title="FINSEQ_3:th.63">FINSEQ_3:63</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:90_2_1_1_1"/>
( ( <font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c2">b1</font> &amp; not <font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c3">b2</font> ) or ( not <font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c2">b1</font> &amp; <font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c3">b2</font> ) or ( <font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c2">b1</font> &amp; <font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c3">b2</font> ) )
 </span><b>by </b><i><a class="txt" href="#E16:90"><i><font color="Green" title="E83">A13</font></i></a>, <a class="txt" href="#E14:90_2_1_1"><i><font color="Green" title="E94">A24</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:90_2_1_1_1_1"/><i><font color="Green" title="E96">A26</font></i>: 
( <font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c2">b1</font> &amp; not <font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c3">b2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c9">f2</font></span>)</span></span>)</span>,<font color="Olive">b<sub>1</sub></font></span>)</span></span><br/><div class="add"><a NAME="E2:90_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E97">A27</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c9">f2</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E12:90"><i><font color="Green" title="E79">A9</font></i></a>, <a class="txt" href="#E13:90"><i><font color="Green" title="E80">A10</font></i></a>, <a class="txt" href="#E14:90"><i><font color="Green" title="E81">A11</font></i></a>, <a class="txt" href="#E4:90_2_1_1"><i><font color="Green" title="E89">A19</font></i></a>, <a class="txt" href="#E9:90_2_1_1"><i><font color="Green" title="E92">A22</font></i></a>, <a class="txt" href="#E11:90_2_1_1"><i><font color="Green" title="E93">A23</font></i></a>, <a class="ref" href="uproots.html#T65" target="_self" title="UPROOTS:th.65">Th65</a></i>;<br/><b>thus </b> <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c15">x</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c3">b2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:90"><i><font color="Green" title="E71">A1</font></i></a>, <a class="txt" href="#E3:90"><i><font color="Green" title="E72">A2</font></i></a>, <a class="txt" href="#E4:90"><i><font color="Green" title="E73">A3</font></i></a>, <a class="txt" href="#E4:90_2_1_1"><i><font color="Green" title="E89">A19</font></i></a>, <a class="txt" href="#E9:90_2_1_1"><i><font color="Green" title="E92">A22</font></i></a>, <a class="txt" href="#E11:90_2_1_1"><i><font color="Green" title="E93">A23</font></i></a>, <a class="txt" href="#E14:90_2_1_1"><i><font color="Green" title="E94">A24</font></i></a>, <a class="ref" href="uproots.html#T65" target="_self" title="UPROOTS:th.65">Th65</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D5" title="POLYNOM1:def.5">POLYNOM1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="#E1:90_2_1_1_1_1"><i><font color="Green" title="E96">A26</font></i></a>, <a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>
<br/>.= 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c9">f2</font></span>)</span></span>)</span>,<font color="Maroon" title="c15">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E7:90"><i><font color="Green" title="E75">A5</font></i></a>, <a class="txt" href="#E8:90"><i><font color="Green" title="E76">A6</font></i></a>, <a class="txt" href="#E9:90"><i><font color="Green" title="E77">A7</font></i></a>, <a class="txt" href="#E4:90_2_1_1"><i><font color="Green" title="E89">A19</font></i></a>, <a class="txt" href="#E9:90_2_1_1"><i><font color="Green" title="E92">A22</font></i></a>, <a class="txt" href="#E11:90_2_1_1"><i><font color="Green" title="E93">A23</font></i></a>, <a class="txt" href="#E15:90_2_1_1"><i><font color="Green" title="E95">A25</font></i></a>, <a class="txt" href="#E1:90_2_1_1_1_1"><i><font color="Green" title="E96">A26</font></i></a>, <a class="txt" href="#E2:90_2_1_1_1_1"><i><font color="Green" title="E97">A27</font></i></a>, <a class="ref" href="uproots.html#T65" target="_self" title="UPROOTS:th.65">Th65</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:90_2_1_1_1_2"/><i><font color="Green" title="E96">A28</font></i>: 
( not <font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c2">b1</font> &amp; <font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c3">b2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c9">f2</font></span>)</span></span>)</span>,<font color="Olive">b<sub>1</sub></font></span>)</span></span><br/><div class="add"><a NAME="E2:90_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E97">A29</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c9">f2</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E12:90"><i><font color="Green" title="E79">A9</font></i></a>, <a class="txt" href="#E13:90"><i><font color="Green" title="E80">A10</font></i></a>, <a class="txt" href="#E14:90"><i><font color="Green" title="E81">A11</font></i></a>, <a class="txt" href="#E4:90_2_1_1"><i><font color="Green" title="E89">A19</font></i></a>, <a class="txt" href="#E9:90_2_1_1"><i><font color="Green" title="E92">A22</font></i></a>, <a class="txt" href="#E11:90_2_1_1"><i><font color="Green" title="E93">A23</font></i></a>, <a class="ref" href="uproots.html#T65" target="_self" title="UPROOTS:th.65">Th65</a></i>;<br/><b>thus </b> <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c15">x</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c3">b2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:90"><i><font color="Green" title="E71">A1</font></i></a>, <a class="txt" href="#E3:90"><i><font color="Green" title="E72">A2</font></i></a>, <a class="txt" href="#E4:90"><i><font color="Green" title="E73">A3</font></i></a>, <a class="txt" href="#E4:90_2_1_1"><i><font color="Green" title="E89">A19</font></i></a>, <a class="txt" href="#E9:90_2_1_1"><i><font color="Green" title="E92">A22</font></i></a>, <a class="txt" href="#E11:90_2_1_1"><i><font color="Green" title="E93">A23</font></i></a>, <a class="txt" href="#E14:90_2_1_1"><i><font color="Green" title="E94">A24</font></i></a>, <a class="ref" href="uproots.html#T65" target="_self" title="UPROOTS:th.65">Th65</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D5" title="POLYNOM1:def.5">POLYNOM1:def 5</a></i>
<br/>.= 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:90_2_1_1_1_2"><i><font color="Green" title="E96">A28</font></i></a>, <a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>
<br/>.= 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c9">f2</font></span>)</span></span>)</span>,<font color="Maroon" title="c15">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E7:90"><i><font color="Green" title="E75">A5</font></i></a>, <a class="txt" href="#E8:90"><i><font color="Green" title="E76">A6</font></i></a>, <a class="txt" href="#E9:90"><i><font color="Green" title="E77">A7</font></i></a>, <a class="txt" href="#E4:90_2_1_1"><i><font color="Green" title="E89">A19</font></i></a>, <a class="txt" href="#E9:90_2_1_1"><i><font color="Green" title="E92">A22</font></i></a>, <a class="txt" href="#E11:90_2_1_1"><i><font color="Green" title="E93">A23</font></i></a>, <a class="txt" href="#E15:90_2_1_1"><i><font color="Green" title="E95">A25</font></i></a>, <a class="txt" href="#E1:90_2_1_1_1_2"><i><font color="Green" title="E96">A28</font></i></a>, <a class="txt" href="#E2:90_2_1_1_1_2"><i><font color="Green" title="E97">A29</font></i></a>, <a class="ref" href="uproots.html#T65" target="_self" title="UPROOTS:th.65">Th65</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_2_1_1_1_3"><b>suppose </b></a><a NAME="E1:90_2_1_1_1_3"/><i><font color="Green" title="E96">A30</font></i>: 
( <font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c2">b1</font> &amp; <font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c3">b2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c9">f2</font></span>)</span></span>)</span>,<font color="Olive">b<sub>1</sub></font></span>)</span></span><br/><div class="add"><a NAME="E2:90_2_1_1_1_3"/><b>then </b><i><font color="Green" title="E97">A31</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c9">f2</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E12:90"><i><font color="Green" title="E79">A9</font></i></a>, <a class="txt" href="#E13:90"><i><font color="Green" title="E80">A10</font></i></a>, <a class="txt" href="#E14:90"><i><font color="Green" title="E81">A11</font></i></a>, <a class="txt" href="#E4:90_2_1_1"><i><font color="Green" title="E89">A19</font></i></a>, <a class="txt" href="#E9:90_2_1_1"><i><font color="Green" title="E92">A22</font></i></a>, <a class="txt" href="#E11:90_2_1_1"><i><font color="Green" title="E93">A23</font></i></a>, <a class="ref" href="uproots.html#T65" target="_self" title="UPROOTS:th.65">Th65</a></i>;<br/><b>thus </b> <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c15">x</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c3">b2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:90"><i><font color="Green" title="E71">A1</font></i></a>, <a class="txt" href="#E3:90"><i><font color="Green" title="E72">A2</font></i></a>, <a class="txt" href="#E4:90"><i><font color="Green" title="E73">A3</font></i></a>, <a class="txt" href="#E4:90_2_1_1"><i><font color="Green" title="E89">A19</font></i></a>, <a class="txt" href="#E9:90_2_1_1"><i><font color="Green" title="E92">A22</font></i></a>, <a class="txt" href="#E11:90_2_1_1"><i><font color="Green" title="E93">A23</font></i></a>, <a class="txt" href="#E14:90_2_1_1"><i><font color="Green" title="E94">A24</font></i></a>, <a class="ref" href="uproots.html#T65" target="_self" title="UPROOTS:th.65">Th65</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D5" title="POLYNOM1:def.5">POLYNOM1:def 5</a></i>
<br/>.= 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c9">f2</font></span>)</span></span>)</span>,<font color="Maroon" title="c15">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E7:90"><i><font color="Green" title="E75">A5</font></i></a>, <a class="txt" href="#E8:90"><i><font color="Green" title="E76">A6</font></i></a>, <a class="txt" href="#E9:90"><i><font color="Green" title="E77">A7</font></i></a>, <a class="txt" href="#E4:90_2_1_1"><i><font color="Green" title="E89">A19</font></i></a>, <a class="txt" href="#E9:90_2_1_1"><i><font color="Green" title="E92">A22</font></i></a>, <a class="txt" href="#E11:90_2_1_1"><i><font color="Green" title="E93">A23</font></i></a>, <a class="txt" href="#E15:90_2_1_1"><i><font color="Green" title="E95">A25</font></i></a>, <a class="txt" href="#E1:90_2_1_1_1_3"><i><font color="Green" title="E96">A30</font></i></a>, <a class="txt" href="#E2:90_2_1_1_1_3"><i><font color="Green" title="E97">A31</font></i></a>, <a class="ref" href="uproots.html#T65" target="_self" title="UPROOTS:th.65">Th65</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_2_1_2"><b>suppose </b></a><a NAME="E1:90_2_1_2"/><i><font color="Green" title="E88">A32</font></i>: 
not <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c9">f2</font></span>)</span></span>)</span>,<font color="Olive">b<sub>1</sub></font></span>)</span></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:90_2_1_2_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c9">f2</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:90_2_1_2_1"/><b>then </b><i><font color="Green" title="E89">A33</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c7">f1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c9">f2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_2_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="E1:90_2_1_2_1_1"/>
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c7">f1</font></span>)</span> or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c9">f2</font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="#E2:90_2_1_2_1"><i><font color="Green" title="E89">A33</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:90_2_1_2_1_1_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c7">f1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c16">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:90_2_1_2_1_1_1"/><i><font color="Green" title="E90">A34</font></i>: 
<font color="Maroon" title="c16">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c7">f1</font></span>)</span>
 <b>and </b><br/><a NAME="E4:90_2_1_2_1_1_1"/><i><font color="Green" title="E91">A35</font></i>: 
<span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/><b>consider </b><font color="Maroon" title="c17">i</font>, <font color="Maroon" title="c18">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:90_2_1_2_1_1_1"/><i><font color="Green" title="E92">A36</font></i>: 
<font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">f1</font>
 <b>and </b><br/><a NAME="E7:90_2_1_2_1_1_1"/><i><font color="Green" title="E93">A37</font></i>: 
<font color="Maroon" title="c18">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font></span>)</span>
 <b>and </b><br/><a NAME="E8:90_2_1_2_1_1_1"/>
<font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K4" title="POLYNOM1:func.4">Card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">j</font>
 <b>and </b><br/><a NAME="E9:90_2_1_2_1_1_1"/><i><font color="Green" title="E94">A38</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font>
 <b>by </b><i><a class="txt" href="#E3:90_2_1_2_1_1_1"><i><font color="Green" title="E90">A34</font></i></a>, <a class="ref" href="polynom1.html#T32" title="POLYNOM1:th.32">POLYNOM1:32</a></i>;<br/><a NAME="E10:90_2_1_2_1_1_1"/>
<font color="Maroon" title="c7">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K9" title="UPROOTS:func.9">fpoly_mult_root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E9:90"><i><font color="Green" title="E77">A7</font></i></a>, <a class="txt" href="#E6:90_2_1_2_1_1_1"><i><font color="Green" title="E92">A36</font></i></a></i>;<br/><a NAME="E11:90_2_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E95">A39</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span>
 <b>by </b><i><a class="txt" href="#E7:90_2_1_2_1_1_1"><i><font color="Green" title="E93">A37</font></i></a>, <a class="ref" href="uproots.html#D10" target="_self" title="UPROOTS:def.10">Def10</a></i>;<br/><a NAME="E12:90_2_1_2_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c2">b1</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E8:90"><i><font color="Green" title="E76">A6</font></i></a>, <a class="ref" href="uproots.html#T5" target="_self" title="UPROOTS:th.5">Def1</a></i>;<br/><a NAME="E13:90_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">s1</font>
 <b>by </b><i><a class="txt" href="#E7:90"><i><font color="Green" title="E75">A5</font></i></a>, <a class="txt" href="#E6:90_2_1_2_1_1_1"><i><font color="Green" title="E92">A36</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E14:90_2_1_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c8">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font> &amp; <font color="Maroon" title="c8">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">s1</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E15:90_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">s1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c2">b1</font>
 <b>by </b><i><a class="txt" href="#E8:90"><i><font color="Green" title="E76">A6</font></i></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/><a NAME="E16:90_2_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E96">A40</font></i>: 
<font color="Maroon" title="c8">s1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c3">b2</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E16:90"><i><font color="Green" title="E83">A13</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E17:90_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">s1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">s</font>
 <b>by </b><i><a class="txt" href="#E3:90"><i><font color="Green" title="E72">A2</font></i></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c19">l</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E19:90_2_1_2_1_1_1"/><i><font color="Green" title="E97">A41</font></i>: 
<font color="Maroon" title="c19">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">s</font>
 <b>and </b><br/><a NAME="E20:90_2_1_2_1_1_1"/><i><font color="Green" title="E98">A42</font></i>: 
<font color="Maroon" title="c6">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/><a NAME="E21:90_2_1_2_1_1_1"/><i><font color="Green" title="E99">A43</font></i>: 
<font color="Maroon" title="c19">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">f</font>
 <b>by </b><i><a class="txt" href="#E2:90"><i><font color="Green" title="E71">A1</font></i></a>, <a class="txt" href="#E21:90"><i><font color="Green" title="E87">A17</font></i></a>, <a class="txt" href="#E19:90_2_1_2_1_1_1"><i><font color="Green" title="E97">A41</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E22:90_2_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E100">A44</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K9" title="UPROOTS:func.9">fpoly_mult_root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">l</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c3">b2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">l</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:90"><i><font color="Green" title="E73">A3</font></i></a></i>;<br/><a NAME="E23:90_2_1_2_1_1_1"/><i><font color="Green" title="E101">A45</font></i>: 
<font color="Maroon" title="c6">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">l</font>
 <b>by </b><i><a class="txt" href="#E19:90_2_1_2_1_1_1"><i><font color="Green" title="E97">A41</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E24:90_2_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c3">b2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E16:90_2_1_2_1_1_1"><i><font color="Green" title="E96">A40</font></i></a>, <a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>;<br/><a NAME="E25:90_2_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E102">A46</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c3">b2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E26:90_2_1_2_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c3">b2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">l</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E22:90_2_1_2_1_1_1"><i><font color="Green" title="E100">A44</font></i></a>, <a class="ref" href="uproots.html#D10" target="_self" title="UPROOTS:def.10">Def10</a></i>;<br/><a NAME="E27:90_2_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E103">A47</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">l</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E20:90_2_1_2_1_1_1"><i><font color="Green" title="E98">A42</font></i></a>, <a class="txt" href="#E23:90_2_1_2_1_1_1"><i><font color="Green" title="E101">A45</font></i></a>, <a class="txt" href="#E25:90_2_1_2_1_1_1"><i><font color="Green" title="E102">A46</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E28:90_2_1_2_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K4" title="POLYNOM1:func.4">Card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">l</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p3">(<span class="default"><a href="polynom1.html#K4" title="POLYNOM1:func.4">Card</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c19">l</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="#E21:90_2_1_2_1_1_1"><i><font color="Green" title="E99">A43</font></i></a>, <a class="ref" href="polynom1.html#T33" title="POLYNOM1:th.33">POLYNOM1:33</a></i>;<br/><a NAME="E29:90_2_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E30:90_2_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:90_2_1_2"><i><font color="Green" title="E88">A32</font></i></a>, <a class="txt" href="#E4:90_2_1_2_1_1_1"><i><font color="Green" title="E91">A35</font></i></a>, <a class="txt" href="#E9:90_2_1_2_1_1_1"><i><font color="Green" title="E94">A38</font></i></a>, <a class="txt" href="#E11:90_2_1_2_1_1_1"><i><font color="Green" title="E95">A39</font></i></a>, <a class="txt" href="#E20:90_2_1_2_1_1_1"><i><font color="Green" title="E98">A42</font></i></a>, <a class="txt" href="#E22:90_2_1_2_1_1_1"><i><font color="Green" title="E100">A44</font></i></a>, <a class="txt" href="#E23:90_2_1_2_1_1_1"><i><font color="Green" title="E101">A45</font></i></a>, <a class="txt" href="#E27:90_2_1_2_1_1_1"><i><font color="Green" title="E103">A47</font></i></a>, <a class="ref" href="uproots.html#D10" target="_self" title="UPROOTS:def.10">Def10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:90_2_1_2_1_1_2"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c9">f2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c16">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:90_2_1_2_1_1_2"/><i><font color="Green" title="E90">A48</font></i>: 
<font color="Maroon" title="c16">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c9">f2</font></span>)</span>
 <b>and </b><br/><a NAME="E4:90_2_1_2_1_1_2"/><i><font color="Green" title="E91">A49</font></i>: 
<span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c9">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/><b>consider </b><font color="Maroon" title="c17">i</font>, <font color="Maroon" title="c18">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:90_2_1_2_1_1_2"/><i><font color="Green" title="E92">A50</font></i>: 
<font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">f2</font>
 <b>and </b><br/><a NAME="E7:90_2_1_2_1_1_2"/><i><font color="Green" title="E93">A51</font></i>: 
<font color="Maroon" title="c18">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font></span>)</span>
 <b>and </b><br/><a NAME="E8:90_2_1_2_1_1_2"/>
<font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K4" title="POLYNOM1:func.4">Card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">j</font>
 <b>and </b><br/><a NAME="E9:90_2_1_2_1_1_2"/><i><font color="Green" title="E94">A52</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c9">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font>
 <b>by </b><i><a class="txt" href="#E3:90_2_1_2_1_1_2"><i><font color="Green" title="E90">A48</font></i></a>, <a class="ref" href="polynom1.html#T32" title="POLYNOM1:th.32">POLYNOM1:32</a></i>;<br/><a NAME="E10:90_2_1_2_1_1_2"/>
<font color="Maroon" title="c9">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K9" title="UPROOTS:func.9">fpoly_mult_root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">s2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">s2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E14:90"><i><font color="Green" title="E81">A11</font></i></a>, <a class="txt" href="#E6:90_2_1_2_1_1_2"><i><font color="Green" title="E92">A50</font></i></a></i>;<br/><a NAME="E11:90_2_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E95">A53</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">s2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span>
 <b>by </b><i><a class="txt" href="#E7:90_2_1_2_1_1_2"><i><font color="Green" title="E93">A51</font></i></a>, <a class="ref" href="uproots.html#D10" target="_self" title="UPROOTS:def.10">Def10</a></i>;<br/><a NAME="E12:90_2_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c3">b2</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E13:90"><i><font color="Green" title="E80">A10</font></i></a>, <a class="ref" href="uproots.html#T5" target="_self" title="UPROOTS:th.5">Def1</a></i>;<br/><a NAME="E13:90_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">s2</font>
 <b>by </b><i><a class="txt" href="#E12:90"><i><font color="Green" title="E79">A9</font></i></a>, <a class="txt" href="#E6:90_2_1_2_1_1_2"><i><font color="Green" title="E92">A50</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E14:90_2_1_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c10">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">s2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font> &amp; <font color="Maroon" title="c10">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">s2</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E15:90_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">s2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c3">b2</font>
 <b>by </b><i><a class="txt" href="#E13:90"><i><font color="Green" title="E80">A10</font></i></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/><a NAME="E16:90_2_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E96">A54</font></i>: 
<font color="Maroon" title="c10">s2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c3">b2</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E16:90"><i><font color="Green" title="E83">A13</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E17:90_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">s2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">s</font>
 <b>by </b><i><a class="txt" href="#E3:90"><i><font color="Green" title="E72">A2</font></i></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c19">l</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E19:90_2_1_2_1_1_2"/><i><font color="Green" title="E97">A55</font></i>: 
<font color="Maroon" title="c19">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">s</font>
 <b>and </b><br/><a NAME="E20:90_2_1_2_1_1_2"/><i><font color="Green" title="E98">A56</font></i>: 
<font color="Maroon" title="c6">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">s2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/><a NAME="E21:90_2_1_2_1_1_2"/><i><font color="Green" title="E99">A57</font></i>: 
<font color="Maroon" title="c19">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">f</font>
 <b>by </b><i><a class="txt" href="#E2:90"><i><font color="Green" title="E71">A1</font></i></a>, <a class="txt" href="#E21:90"><i><font color="Green" title="E87">A17</font></i></a>, <a class="txt" href="#E19:90_2_1_2_1_1_2"><i><font color="Green" title="E97">A55</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E22:90_2_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E100">A58</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K9" title="UPROOTS:func.9">fpoly_mult_root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">l</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c3">b2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">l</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:90"><i><font color="Green" title="E73">A3</font></i></a></i>;<br/><a NAME="E23:90_2_1_2_1_1_2"/><i><font color="Green" title="E101">A59</font></i>: 
<font color="Maroon" title="c6">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">l</font>
 <b>by </b><i><a class="txt" href="#E19:90_2_1_2_1_1_2"><i><font color="Green" title="E97">A55</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E24:90_2_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c3">b2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">s2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E16:90_2_1_2_1_1_2"><i><font color="Green" title="E96">A54</font></i></a>, <a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>;<br/><a NAME="E25:90_2_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E102">A60</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c3">b2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">s2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E26:90_2_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c3">b2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">l</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E22:90_2_1_2_1_1_2"><i><font color="Green" title="E100">A58</font></i></a>, <a class="ref" href="uproots.html#D10" target="_self" title="UPROOTS:def.10">Def10</a></i>;<br/><a NAME="E27:90_2_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E103">A61</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">l</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E20:90_2_1_2_1_1_2"><i><font color="Green" title="E98">A56</font></i></a>, <a class="txt" href="#E23:90_2_1_2_1_1_2"><i><font color="Green" title="E101">A59</font></i></a>, <a class="txt" href="#E25:90_2_1_2_1_1_2"><i><font color="Green" title="E102">A60</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E28:90_2_1_2_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K4" title="POLYNOM1:func.4">Card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">l</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p3">(<span class="default"><a href="polynom1.html#K4" title="POLYNOM1:func.4">Card</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c19">l</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="#E21:90_2_1_2_1_1_2"><i><font color="Green" title="E99">A57</font></i></a>, <a class="ref" href="polynom1.html#T33" title="POLYNOM1:th.33">POLYNOM1:33</a></i>;<br/><a NAME="E29:90_2_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E30:90_2_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:90_2_1_2"><i><font color="Green" title="E88">A32</font></i></a>, <a class="txt" href="#E4:90_2_1_2_1_1_2"><i><font color="Green" title="E91">A49</font></i></a>, <a class="txt" href="#E9:90_2_1_2_1_1_2"><i><font color="Green" title="E94">A52</font></i></a>, <a class="txt" href="#E11:90_2_1_2_1_1_2"><i><font color="Green" title="E95">A53</font></i></a>, <a class="txt" href="#E20:90_2_1_2_1_1_2"><i><font color="Green" title="E98">A56</font></i></a>, <a class="txt" href="#E22:90_2_1_2_1_1_2"><i><font color="Green" title="E100">A58</font></i></a>, <a class="txt" href="#E23:90_2_1_2_1_1_2"><i><font color="Green" title="E101">A59</font></i></a>, <a class="txt" href="#E27:90_2_1_2_1_1_2"><i><font color="Green" title="E103">A61</font></i></a>, <a class="ref" href="uproots.html#D10" target="_self" title="UPROOTS:def.10">Def10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E3:90_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c9">f2</font></span>)</span></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="funct_1.html#T142" title="FUNCT_1:th.142">FUNCT_1:142</a></i>;<br/><b>hence </b><a NAME="E4:90_2_1_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c15">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c9">f2</font></span>)</span></span>)</span>,<font color="Maroon" title="c15">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:90_2_1_2"><i><font color="Green" title="E88">A32</font></i></a>, <a class="ref" href="funct_1.html#T142" title="FUNCT_1:th.142">FUNCT_1:142</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E23:90"/><b>then </b>
 <a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">f</font>,<span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c9">f2</font></span>)</span> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="ref" href="classes1.html#D9" title="CLASSES1:def.9">CLASSES1:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">p</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">f</font></span>)</span><b> such that </b><br/><a NAME="E25:90"/><i><font color="Green" title="E88">A62</font></i>: 
 <a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c9">f2</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c15">p</font>
 <b>by </b><i><a class="txt" href="#E20:90"><i><font color="Green" title="E86">A16</font></i></a>, <a class="ref" href="rfinseq.html#T17" title="RFINSEQ:th.17">RFINSEQ:17</a></i>;<br/>
<b>thus </b> <a href="uproots.html#K10" title="UPROOTS:func.10">poly_with_roots</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c3">b2</font></span>)</span> = 
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c9">f2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E5:90"><i><font color="Green" title="E74">A4</font></i></a>, <a class="txt" href="#E20:90"><i><font color="Green" title="E86">A16</font></i></a>, <a class="txt" href="#E25:90"><i><font color="Green" title="E88">A62</font></i></a>, <a class="ref" href="uproots.html#T18" target="_self" title="UPROOTS:th.18">Th18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c7">f1</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c9">f2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_4.html#T8" title="GROUP_4:th.8">GROUP_4:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="uproots.html#K10" title="UPROOTS:func.10">poly_with_roots</a> <font color="Maroon" title="c2">b1</font></span>)</span> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <span class="p1">(<span class="default"><a href="uproots.html#K10" title="UPROOTS:func.10">poly_with_roots</a> <font color="Maroon" title="c3">b2</font></span>)</span>
<b>by </b><i><a class="txt" href="#E10:90"><i><font color="Green" title="E78">A8</font></i></a>, <a class="txt" href="#E15:90"><i><font color="Green" title="E82">A12</font></i></a>, <a class="ref" href="polynom3.html#D12" title="POLYNOM3:def.12">POLYNOM3:def 12</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
