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<a NAME="E1:51"/>
1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<a NAME="E2:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A1</font></span>: 
1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c1">z2</font> =  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#D1" title="COMPLFLD:def.1">COMPLFLD:def 1</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c2">n</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> holds   <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> (<a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Olive" title="b1">x</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="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="binop_2.html#K4" title="BINOP_2:func.4">-</a> 1</span><br/><span class="kw">let </span><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> (<a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c3">x</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="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="binop_2.html#K4" title="BINOP_2:func.4">-</a> 1</span><br/>



<span class="kw">set </span><font color="Maroon" title="c4">p</font> =  <a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> (<a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>,<font color="Maroon" title="c2">n</font>);<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">F</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a><span class="kw"> such that </span><br/><a NAME="E5:51"/><span class="lab"><font color="Green" title="E43">A2</font></span>: 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> (<a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c3">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c5">F</font>
 <span class="kw">and </span><br/><a NAME="E6:51"/><span class="lab"><font color="Green" title="E44">A3</font></span>: 
 <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="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> (<a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:51"/><span class="lab"><font color="Green" title="E45">A4</font></span>: 
 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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> (<a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom4.html#D2" title="POLYNOM4:def.2">POLYNOM4:def 2</a></span>;<br/>
<a NAME="E8:51"/><span class="lab"><font color="Green" title="E46">A5</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E9:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A6</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="ref" href="uniroots.html#T40" target="_self" title="UNIROOTS:th.40">Th44</a></span>;<br/>
<a NAME="E10:51"/><span class="lab"><font color="Green" title="E48">A7</font></span>: 
 <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> <font color="Maroon" title="c2">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="ref" href="uniroots.html#T40" target="_self" title="UNIROOTS:th.40">Th44</a></span>;<br/>
<a NAME="E11:51"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 
;<br/>
<a NAME="E12:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A8</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:51"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/>
<a NAME="E13:51"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font>
 
;<br/>
<a NAME="E14:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A9</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:51"><span class="lab"><font color="Green" title="E47">A6</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/>
<a NAME="E15:51"/><span class="lab"><font color="Green" title="E51">A10</font></span>: 
<span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> (<a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="uniroots.html#T38" target="_self" title="UNIROOTS:th.38">Th42</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">xn</font> = <span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">n</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c7">null</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span>;<br/>
<a NAME="E16:51"/><span class="lab"><font color="Green" title="E52">A11</font></span>: 
 <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"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">tR2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">tR1</font> = <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&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"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span></span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span></span>)</span>;<br/>
<a NAME="E17:51"/><span class="lab"><font color="Green" title="E53">A12</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">R1</font> = <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&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"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span></span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span></span>)</span> as   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:51"><span class="lab"><font color="Green" title="E50">A9</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">R1</font> = <font color="Maroon" title="c10">R1</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T131" title="FINSEQ_2:th.131">FINSEQ_2:131</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">R2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> as   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:51"><span class="lab"><font color="Green" title="E50">A9</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">R2</font> = <font color="Maroon" title="c12">R2</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T131" title="FINSEQ_2:th.131">FINSEQ_2:131</a></span>;<br/>
<a NAME="E22:51"/><span class="lab"><font color="Green" title="E54">A13</font></span>: 
 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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span></span>)</span> <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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">i</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> )</span><br/>

<span class="kw">assume </span><a NAME="E1:51_1"/>
<font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span><br/>

<a NAME="E2:51_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:51_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E23:51"/><span class="lab"><font color="Green" title="E55">A14</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <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="c11">R1</font> holds <br/><font color="Maroon" title="c11">R1</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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c14">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="c11">R1</font> implies <font color="Maroon" title="c11">R1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:51_2"/><span class="lab"><font color="Green" title="E56">A15</font></span>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <span class="kw">and </span><br/><a NAME="E2:51_2"/><span class="lab"><font color="Green" title="E57">A16</font></span>: 
<font color="Maroon" title="c14">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="c11">R1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">R1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span><br/>

<a NAME="E3:51_2"/><span class="lab"><font color="Green" title="E58">A17</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&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"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span></span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:51_2_1"/>
<font color="Maroon" title="c14">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> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&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"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span></span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:51_2_1"/><span class="kw">then </span>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_2"><span class="lab"><font color="Green" title="E58">A17</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:51_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2"><span class="lab"><font color="Green" title="E56">A15</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:51_2"/><span class="lab"><font color="Green" title="E59">A18</font></span>: 
<font color="Maroon" title="c15">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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:51_2"/><span class="lab"><font color="Green" title="E60">A19</font></span>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span></span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c15">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_2"><span class="lab"><font color="Green" title="E57">A16</font></span></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></span>;<br/>
<a NAME="E8:51_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:51"><span class="lab"><font color="Green" title="E54">A13</font></span></a>, <a class="txt" href="#E6:51_2"><span class="lab"><font color="Green" title="E59">A18</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:51_2"/>
<font color="Maroon" title="c11">R1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51_2"><span class="lab"><font color="Green" title="E59">A18</font></span></a>, <a class="txt" href="#E7:51_2"><span class="lab"><font color="Green" title="E60">A19</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E24:51"/>
 <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>;<br/>
<a NAME="E25:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A20</font></span>: 
 <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:51"><span class="lab"><font color="Green" title="E52">A11</font></span></a>, <a class="txt" href="#E14:51"><span class="lab"><font color="Green" title="E50">A9</font></span></a>, <a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a></span>;<br/>
<a NAME="E26:51"/><span class="lab"><font color="Green" title="E57">A21</font></span>: 
 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="c13">R2</font> &amp; <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font> holds <br/><font color="Maroon" title="c13">R2</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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">i</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">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="c13">R2</font> &amp; <font color="Maroon" title="c14">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font> implies <font color="Maroon" title="c13">R2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:51_3"/><span class="lab"><font color="Green" title="E58">A22</font></span>: 
<font color="Maroon" title="c14">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="c13">R2</font>
 <span class="kw">and </span><br/><a NAME="E2:51_3"/><span class="lab"><font color="Green" title="E59">A23</font></span>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">R2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span><br/>

<a NAME="E3:51_3"/><span class="lab"><font color="Green" title="E60">A24</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">R2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:51"><span class="lab"><font color="Green" title="E56">A20</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E4:51_3"/><span class="kw">then </span>
<font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_3"><span class="lab"><font color="Green" title="E58">A22</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E5:51_3"/><span class="kw">then </span>
<font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:51"><span class="lab"><font color="Green" title="E50">A9</font></span></a>, <a class="txt" href="#E2:51_3"><span class="lab"><font color="Green" title="E59">A23</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E6:51_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A25</font></span>: 
<font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E7:51_3"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_3"><span class="lab"><font color="Green" title="E58">A22</font></span></a>, <a class="txt" href="#E3:51_3"><span class="lab"><font color="Green" title="E60">A24</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E8:51_3"/><span class="kw">then </span>
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51_3"><span class="lab"><font color="Green" title="E61">A25</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E9:51_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A26</font></span>: 
<font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:51"><span class="lab"><font color="Green" title="E52">A11</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E10:51_3"/><span class="kw">then </span>
<font color="Maroon" title="c13">R2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:51_3"/>
<font color="Maroon" title="c13">R2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:51"><span class="lab"><font color="Green" title="E54">A13</font></span></a>, <a class="txt" href="#E9:51_3"><span class="lab"><font color="Green" title="E62">A26</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E27:51"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<a NAME="E28:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A27</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E29:51"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">R2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<a NAME="E30:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A28</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">R2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E31:51"/><span class="lab"><font color="Green" title="E60">A29</font></span>: 
<font color="Maroon" title="c11">R1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T41" title="FINSEQ_1:th.41">FINSEQ_1:41</a></span>;<br/>
<a NAME="E32:51"/><span class="lab"><font color="Green" title="E61">A30</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">R1</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c13">R2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<a NAME="E33:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A31</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">R1</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c13">R2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E34:51"/><span class="lab"><font color="Green" title="E63">A32</font></span>: 
<font color="Maroon" title="c13">R2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:51"><span class="lab"><font color="Green" title="E52">A11</font></span></a>, <a class="txt" href="#E14:51"><span class="lab"><font color="Green" title="E50">A9</font></span></a>, <a class="ref" href="finseq_1.html#T42" title="FINSEQ_1:th.42">FINSEQ_1:42</a></span>;<br/>
<a NAME="E35:51"/>
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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"><font color="Maroon" title="c11">R1</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c13">R2</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c11">R1</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c13">R2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">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"><font color="Maroon" title="c11">R1</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c13">R2</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c11">R1</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c13">R2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:51_4"/><span class="lab"><font color="Green" title="E64">A33</font></span>: 
<font color="Maroon" title="c14">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"><font color="Maroon" title="c11">R1</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c13">R2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c11">R1</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c13">R2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font></span><br/>

<a NAME="E2:51_4"/><span class="lab"><font color="Green" title="E65">A34</font></span>: 
<font color="Maroon" title="c14">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:51"><span class="lab"><font color="Green" title="E61">A30</font></span></a>, <a class="txt" href="#E1:51_4"><span class="lab"><font color="Green" title="E64">A33</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E3:51_4"/><span class="lab"><font color="Green" title="E66">A35</font></span>: 
<font color="Maroon" title="c14">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:51"><span class="lab"><font color="Green" title="E62">A31</font></span></a>, <a class="txt" href="#E1:51_4"><span class="lab"><font color="Green" title="E64">A33</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E4:51_4"/><span class="lab"><font color="Green" title="E67">A36</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:51"><span class="lab"><font color="Green" title="E62">A31</font></span></a>, <a class="txt" href="#E1:51_4"><span class="lab"><font color="Green" title="E64">A33</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E5:51_4"/><span class="lab"><font color="Green" title="E68">A37</font></span>: 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></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_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#T8" title="COMPLFLD:th.8">COMPLFLD:8</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_4_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_4_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_4_1_1"/>
( <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c14">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_4_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:51_4_1_1_1"/><span class="lab"><font color="Green" title="E69">A38</font></span>: 
<font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c11">R1</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c13">R2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font></span><br/><div class="add"><a NAME="E2:51_4_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">R2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:51"><span class="lab"><font color="Green" title="E47">A6</font></span></a>, <a class="txt" href="#E26:51"><span class="lab"><font color="Green" title="E57">A21</font></span></a>, <a class="txt" href="#E30:51"><span class="lab"><font color="Green" title="E59">A28</font></span></a>, <a class="txt" href="#E2:51_4"><span class="lab"><font color="Green" title="E65">A34</font></span></a></span>;<br/><a NAME="E3:51_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A39</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">R1</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c13">R2</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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:51"><span class="lab"><font color="Green" title="E60">A29</font></span></a>, <a class="txt" href="#E1:51_4"><span class="lab"><font color="Green" title="E64">A33</font></span></a>, <a class="txt" href="#E1:51_4_1_1_1"><span class="lab"><font color="Green" title="E69">A38</font></span></a>, <a class="ref" href="fvsum_1.html#T17" title="FVSUM_1:th.17">FVSUM_1:17</a></span>;<br/><font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> (<a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">x</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:51"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="txt" href="#E2:51"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="txt" href="#E3:51_4"><span class="lab"><font color="Green" title="E66">A35</font></span></a>, <a class="txt" href="#E1:51_4_1_1_1"><span class="lab"><font color="Green" title="E69">A38</font></span></a></span>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:51"><span class="lab"><font color="Green" title="E51">A10</font></span></a>, <a class="txt" href="#E5:51_4"><span class="lab"><font color="Green" title="E68">A37</font></span></a>, <a class="ref" href="group_1.html#D7" title="GROUP_1:def.7">GROUP_1:def 7</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:51_4_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">R1</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c13">R2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_4_1_1_1"><span class="lab"><font color="Green" title="E69">A38</font></span></a>, <a class="txt" href="#E3:51_4_1_1_1"><span class="lab"><font color="Green" title="E70">A39</font></span></a>, <a class="ref" href="complfld.html#T7" title="COMPLFLD:th.7">COMPLFLD:7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_4_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:51_4_1_1_2"/><span class="lab"><font color="Green" title="E69">A40</font></span>: 
<font color="Maroon" title="c14">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c11">R1</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c13">R2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_4_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_4_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_4_1_1_2_1_1"/>
( <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font> or <font color="Maroon" title="c14">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_4_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:51_4_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E70">A41</font></span>: 
<font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c11">R1</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c13">R2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font></span><br/><div class="add"><a NAME="E2:51_4_1_1_2_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="txt" href="#E8:51"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="ref" href="uniroots.html#T40" target="_self" title="UNIROOTS:th.40">Th44</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A42</font></span>: <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> (<a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:51"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="txt" href="#E17:51"><span class="lab"><font color="Green" title="E53">A12</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:51"><span class="lab"><font color="Green" title="E49">A8</font></span></a>, <a class="ref" href="uniroots.html#T38" target="_self" title="UNIROOTS:th.38">Th42</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">n</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D8" title="VECTSP_1:def.8">VECTSP_1:def 8</a></span>
;<br/><a NAME="E4:51_4_1_1_2_1_1_1"/>
<font color="Maroon" title="c11">R1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:51"><span class="lab"><font color="Green" title="E47">A6</font></span></a>, <a class="txt" href="#E23:51"><span class="lab"><font color="Green" title="E55">A14</font></span></a>, <a class="txt" href="#E28:51"><span class="lab"><font color="Green" title="E58">A27</font></span></a>, <a class="txt" href="#E2:51_4"><span class="lab"><font color="Green" title="E65">A34</font></span></a>, <a class="txt" href="#E1:51_4_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E70">A41</font></span></a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c11">R1</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c13">R2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:51"><span class="lab"><font color="Green" title="E63">A32</font></span></a>, <a class="txt" href="#E1:51_4"><span class="lab"><font color="Green" title="E64">A33</font></span></a>, <a class="txt" href="#E1:51_4_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E70">A41</font></span></a>, <a class="ref" href="fvsum_1.html#T17" title="FVSUM_1:th.17">FVSUM_1:17</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">n</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#T7" title="COMPLFLD:th.7">COMPLFLD:7</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:51_4_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">R1</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c13">R2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_4_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E70">A41</font></span></a>, <a class="txt" href="#E3:51_4_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A42</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_4_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:51_4_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E70">A43</font></span>: 
<font color="Maroon" title="c14">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c11">R1</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c13">R2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font></span><br/><div class="add"><div><span class="lab"><font color="Green" title="E71">A44</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="51_4_1_1_2_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:51_4_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c14">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:51_4_1_1_2_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">k</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_4"><span class="lab"><font color="Green" title="E67">A36</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/><span class="kw">hence </span><a NAME="E3:51_4_1_1_2_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:51"><span class="lab"><font color="Green" title="E48">A7</font></span></a>, <a class="txt" href="#E1:51_4_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E70">A43</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E3:51_4_1_1_2_1_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_4"><span class="lab"><font color="Green" title="E67">A36</font></span></a>, <a class="txt" href="#E1:51_4_1_1_2"><span class="lab"><font color="Green" title="E69">A40</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:51_4_1_1_2_1_1_2"/><span class="kw">then </span>
1 <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p1">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">k</font> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p1">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E5:51_4_1_1_2_1_1_2"/><span class="kw">then </span>
1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">k</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1
 ;<br/><a NAME="E6:51_4_1_1_2_1_1_2"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_4"><span class="lab"><font color="Green" title="E67">A36</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/><a NAME="E7:51_4_1_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> (<a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_4_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E71">A44</font></span></a>, <a class="ref" href="uniroots.html#T39" target="_self" title="UNIROOTS:th.39">Th43</a></span>;<br/><a NAME="E8:51_4_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A45</font></span>: 
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:51"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="txt" href="#E3:51_4"><span class="lab"><font color="Green" title="E66">A35</font></span></a></span>;<br/><a NAME="E9:51_4_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E73">A46</font></span>: 
<font color="Maroon" title="c13">R2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:51"><span class="lab"><font color="Green" title="E57">A21</font></span></a>, <a class="txt" href="#E30:51"><span class="lab"><font color="Green" title="E59">A28</font></span></a>, <a class="txt" href="#E2:51_4"><span class="lab"><font color="Green" title="E65">A34</font></span></a>, <a class="txt" href="#E1:51_4_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E70">A43</font></span></a></span>;<br/><a NAME="E10:51_4_1_1_2_1_1_2"/>
<font color="Maroon" title="c11">R1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:51"><span class="lab"><font color="Green" title="E55">A14</font></span></a>, <a class="txt" href="#E28:51"><span class="lab"><font color="Green" title="E58">A27</font></span></a>, <a class="txt" href="#E2:51_4"><span class="lab"><font color="Green" title="E65">A34</font></span></a>, <a class="txt" href="#E1:51_4_1_1_2"><span class="lab"><font color="Green" title="E69">A40</font></span></a></span>;<br/><a NAME="E11:51_4_1_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">R1</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c13">R2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_4"><span class="lab"><font color="Green" title="E64">A33</font></span></a>, <a class="txt" href="#E9:51_4_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A46</font></span></a>, <a class="ref" href="fvsum_1.html#T17" title="FVSUM_1:th.17">FVSUM_1:17</a></span>;<br/><span class="kw">hence </span><a NAME="E12:51_4_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">R1</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c13">R2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:51_4_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E72">A45</font></span></a>, <a class="ref" href="complfld.html#T7" title="COMPLFLD:th.7">COMPLFLD:7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:51_4_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">R1</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c13">R2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:51_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">R1</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c13">R2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E36:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A47</font></span>: 
<font color="Maroon" title="c11">R1</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c13">R2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:51"><span class="lab"><font color="Green" title="E53">A12</font></span></a>, <a class="txt" href="#E33:51"><span class="lab"><font color="Green" title="E62">A31</font></span></a>, <a class="ref" href="finseq_1.html#T13" title="FINSEQ_1:th.13">FINSEQ_1:13</a></span>;<br/>
<a NAME="E37:51"/>
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_3.html#T11" title="MATRIX_3:th.11">MATRIX_3:11</a></span>;<br/>
<a NAME="E38:51"/><span class="kw">then </span>
(  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c11">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> &amp;  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c13">R2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fvsum_1.html#T71" title="FVSUM_1:th.71">FVSUM_1:71</a>, <a class="ref" href="fvsum_1.html#T72" title="FVSUM_1:th.72">FVSUM_1:72</a></span>;<br/>
<a NAME="E39:51"/><span class="kw">then </span>
(  <a href="binop_2.html#K1" title="BINOP_2:func.1">-</a> <font color="Maroon" title="c1">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> &amp;  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:51"><span class="lab"><font color="Green" title="E64">A47</font></span></a>, <a class="ref" href="complfld.html#T2" title="COMPLFLD:th.2">COMPLFLD:2</a>, <a class="ref" href="complfld.html#T7" title="COMPLFLD:th.7">COMPLFLD:7</a>, <a class="ref" href="fvsum_1.html#T76" title="FVSUM_1:th.76">FVSUM_1:76</a></span>;<br/>
<span class="kw">hence </span><a NAME="E40:51"/>
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> (<a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c3">x</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="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="binop_2.html#K4" title="BINOP_2:func.4">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="ref" href="complfld.html#T8" title="COMPLFLD:th.8">COMPLFLD:8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
