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<span class="kw">let </span><font color="Maroon" title="c2">E</font> be   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c1">F</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">E</font> is <font color="Maroon" title="c1">F</font> <a href="field_4.html#V2" title="FIELD_4:attr.2">-finite</a>  implies <font color="Maroon" title="c2">E</font> is <font color="Maroon" title="c1">F</font> <a href="field_7.html#V2" title="FIELD_7:attr.2">-algebraic</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:73_1_1"/><span class="lab"><font color="Green" title="E42">AS</font></span>: 
<font color="Maroon" title="c2">E</font> is <font color="Maroon" title="c1">F</font> <a href="field_4.html#V2" title="FIELD_4:attr.2">-finite</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">E</font> is <font color="Maroon" title="c1">F</font> <a href="field_7.html#V2" title="FIELD_7:attr.2">-algebraic</a> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">n</font> =  <a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c1">F</font>) as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E3:73_1_1"/><span class="lab"><font color="Green" title="E43">H</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c1">F</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#D7" onmouseover="rs('field_4/D7')" onmouseout="rh()">FIELD_4:def 7</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="73_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> holds  <font color="Olive" title="b1">a</font> is <font color="Maroon" title="c1">F</font> <a href="field_6.html#V2" title="FIELD_6:attr.2">-algebraic</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> is <font color="Maroon" title="c1">F</font> <a href="field_6.html#V2" title="FIELD_6:attr.2">-algebraic</a> </span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:73_1_1_1_1"/>
(  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b2">j</font> ) or  for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   holds <br/>(  not <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> or  not <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> or  not <font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b2">j</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:73_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b2">j</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> is <font color="Maroon" title="c1">F</font> <a href="field_6.html#V2" title="FIELD_6:attr.2">-algebraic</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">i</font>, <font color="Maroon" title="c6">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:73_1_1_1_1_1"/><span class="lab"><font color="Green" title="E44">U</font></span>: 
( <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">j</font> )
 ;<br/><span class="kw">set </span><font color="Maroon" title="c7">p1</font> = <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">`^</a> <font color="Maroon" title="c6">j</font>;<br/><span class="kw">set </span><font color="Maroon" title="c8">p2</font> = <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">`^</a> <font color="Maroon" title="c5">i</font>;<br/><span class="kw">set </span><font color="Maroon" title="c9">p</font> = <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">`^</a> <font color="Maroon" title="c6">j</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">`^</a> <font color="Maroon" title="c5">i</font></span>)</span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="73_1_1_1_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">`^</a> <font color="Maroon" title="c6">j</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">`^</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <font color="Maroon" title="c1">F</font></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:73_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">`^</a> <font color="Maroon" title="c6">j</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">`^</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <font color="Maroon" title="c1">F</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span> <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">`^</a> <font color="Maroon" title="c6">j</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">`^</a> <font color="Maroon" title="c5">i</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">j</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">`^</a> <font color="Maroon" title="c6">j</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">j</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">`^</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#T27" onmouseover="rs('polynom3/T27')" onmouseout="rh()">POLYNOM3:27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">`^</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="field_7.html#E23"><span class="lab"><font color="Green" title="E17">help1</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">U</font></span></a>, <a class="txt" href="field_7.html#E24"><span class="lab"><font color="Green" title="E18">help2</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E3:73_1_1_1_1_1_1"/>
contradiction
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">reconsider </span><font color="Maroon" title="c10">p</font> = <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">`^</a> <font color="Maroon" title="c6">j</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">`^</a> <font color="Maroon" title="c5">i</font></span>)</span> as   non  <a href="ratfunc1.html#NV1" title="RATFUNC1:NV.1">zero</a>  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">F</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#D5" onmouseover="rs('uproots/D5')" onmouseout="rh()">UPROOTS:def 5</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1_1_1_1_2"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:73_1_1_1_1_1_2"/>
( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or  not <font color="Maroon" title="c5">i</font> is  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1_1_1_1_2_1"><span class="kw">suppose </span></a><a NAME="E1:73_1_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E45">T</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> is <font color="Maroon" title="c1">F</font> <a href="field_6.html#V2" title="FIELD_6:attr.2">-algebraic</a> </span><br/><div class="add"><a NAME="E2:73_1_1_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">W</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="binom.html#T8" onmouseover="rs('binom/T8')" onmouseout="rh()">BINOM:8</a></span>;<br/> <a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<font color="Maroon" title="c10">p</font>,<font color="Maroon" title="c4">a</font>) = 
<span class="p1">(<span class="default"><a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<span class="p2">(<span class="default"><span class="p3"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">`^</a> <font color="Maroon" title="c6">j</font></span>)</span>,<font color="Maroon" title="c4">a</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<span class="p2">(<span class="default"><span class="p3"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">`^</a> <font color="Maroon" title="c5">i</font></span>)</span>,<font color="Maroon" title="c4">a</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_6.html#T27" onmouseover="rs('field_6/T27')" onmouseout="rh()">FIELD_6:27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<span class="p2">(<span class="default"><span class="p3"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">`^</a> <font color="Maroon" title="c6">j</font></span>)</span>,<font color="Maroon" title="c4">a</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<span class="p2">(<span class="default"><a href="polynom3.html#K9" title="POLYNOM3:func.9">1_.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<font color="Maroon" title="c4">a</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E45">T</font></span></a>, <a class="ref" href="polynom5.html#T15" onmouseover="rs('polynom5/T15')" onmouseout="rh()">POLYNOM5:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<span class="p2">(<span class="default"><span class="p3"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">`^</a> <font color="Maroon" title="c6">j</font></span>)</span>,<font color="Maroon" title="c4">a</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">E</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#T23" onmouseover="rs('field_4/T23')" onmouseout="rh()">FIELD_4:23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">j</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">E</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">U</font></span></a>, <a class="ref" href="field_6.html#T22" onmouseover="rs('field_6/T22')" onmouseout="rh()">FIELD_6:22</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">E</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E46">W</font></span></a>, <a class="txt" href="#E3:73_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">U</font></span></a>, <a class="ref" href="rlvect_1.html#T15" onmouseover="rs('rlvect_1/T15')" onmouseout="rh()">RLVECT_1:15</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:73_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c4">a</font> is <font color="Maroon" title="c1">F</font> <a href="field_6.html#V2" title="FIELD_6:attr.2">-algebraic</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_6.html#T43" onmouseover="rs('field_6/T43')" onmouseout="rh()">FIELD_6:43</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="73_1_1_1_1_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:73_1_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E45">T</font></span>: 
 not <font color="Maroon" title="c5">i</font> is  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> is <font color="Maroon" title="c1">F</font> <a href="field_6.html#V2" title="FIELD_6:attr.2">-algebraic</a> </span><br/><div class="add"> <a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<font color="Maroon" title="c10">p</font>,<font color="Maroon" title="c4">a</font>) = 
<span class="p1">(<span class="default"><a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<span class="p2">(<span class="default"><span class="p3"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">`^</a> <font color="Maroon" title="c6">j</font></span>)</span>,<font color="Maroon" title="c4">a</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<span class="p2">(<span class="default"><span class="p3"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">`^</a> <font color="Maroon" title="c5">i</font></span>)</span>,<font color="Maroon" title="c4">a</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_6.html#T27" onmouseover="rs('field_6/T27')" onmouseout="rh()">FIELD_6:27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">j</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<span class="p2">(<span class="default"><span class="p3"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">`^</a> <font color="Maroon" title="c5">i</font></span>)</span>,<font color="Maroon" title="c4">a</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">U</font></span></a>, <a class="ref" href="field_6.html#T22" onmouseover="rs('field_6/T22')" onmouseout="rh()">FIELD_6:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">j</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c5">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_1_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E45">T</font></span></a>, <a class="ref" href="field_6.html#T22" onmouseover="rs('field_6/T22')" onmouseout="rh()">FIELD_6:22</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">E</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">U</font></span></a>, <a class="ref" href="rlvect_1.html#T15" onmouseover="rs('rlvect_1/T15')" onmouseout="rh()">RLVECT_1:15</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:73_1_1_1_1_1_2_2"/>
<font color="Maroon" title="c4">a</font> is <font color="Maroon" title="c1">F</font> <a href="field_6.html#V2" title="FIELD_6:attr.2">-algebraic</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_6.html#T43" onmouseover="rs('field_6/T43')" onmouseout="rh()">FIELD_6:43</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:73_1_1_1_1_2"/><span class="lab"><font color="Green" title="E44">U</font></span>: 
 for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   holds <br/>(  not <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> or  not <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> or  not <font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b2">j</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> is <font color="Maroon" title="c1">F</font> <a href="field_6.html#V2" title="FIELD_6:attr.2">-algebraic</a> </span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c5">M</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> </span> } </span> ;<br/><span class="kw">set </span><font color="Maroon" title="c6">V</font> =  <a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c1">F</font>);<br/><a NAME="E2:73_1_1_1_1_2"/><span class="lab"><font color="Green" title="E45">I</font></span>: 
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c1">F</font>)</span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="73_1_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="73_1_1_1_1_2_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b2">i</font></span>)</span> where <font color="Olive" title="b2">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> </span> } </span>  holds <br/><font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> </span> } </span>  implies <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c1">F</font>)</span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:73_1_1_1_1_2_1_1"/>
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> </span> } </span> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:73_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E46">H</font></span>: 
( <font color="Maroon" title="c7">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> )
 ;<br/><a NAME="E4:73_1_1_1_1_2_1_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c1">F</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#D6" onmouseover="rs('field_4/D6')" onmouseout="rh()">FIELD_4:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E5:73_1_1_1_1_2_1_1"/>
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c1">F</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E46">H</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">hence </span><a NAME="E2:73_1_1_1_1_2_1"/>
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c1">F</font>)</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E3:73_1_1_1_1_2"/>
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> </span> } </span>  is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="73_1_1_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">E</font> = <font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c1">F</font>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span><font color="Maroon" title="c1">F</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font>;<br/>
<a NAME="E1:73_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E46">D</font></span>: 
<span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] ) </span> } </span>  is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S6" onmouseover="rs('finseq_1/S6')" onmouseout="rh()">FINSEQ_1:sch 6</a>();<br/></span>
<div><span class="lab"><font color="Green" title="E47">E</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="73_1_1_1_1_2_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">i</font>) where <font color="Olive" title="b2">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">i</font>] ) </span> } </span>  holds <br/><font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b2">i</font></span>)</span> where <font color="Olive" title="b2">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> </span> } </span> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] ) </span> } </span>  implies <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> </span> } </span>  )</span><br/><span class="kw">assume </span><a NAME="E1:73_1_1_1_1_2_2_1"/>
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] ) </span> } </span> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> </span> } </span> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:73_1_1_1_1_2_2_1"/><span class="lab"><font color="Green" title="E48">E1</font></span>: 
( <font color="Maroon" title="c7">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> )
 ;<br/><a NAME="E4:73_1_1_1_1_2_2_1"/>
<font color="Maroon" title="c8">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/><span class="kw">hence </span><a NAME="E5:73_1_1_1_1_2_2_1"/>
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_1_1_1_2_2_1"><span class="lab"><font color="Green" title="E48">E1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="73_1_1_1_1_2_2_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b2">i</font></span>)</span> where <font color="Olive" title="b2">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> </span> } </span>  holds <br/><font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">i</font>) where <font color="Olive" title="b2">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">i</font>] ) </span> } </span> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> </span> } </span>  implies <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] ) </span> } </span>  )</span><br/><span class="kw">assume </span><a NAME="E1:73_1_1_1_1_2_2_2"/>
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> </span> } </span> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] ) </span> } </span> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:73_1_1_1_1_2_2_2"/><span class="lab"><font color="Green" title="E48">E1</font></span>: 
( <font color="Maroon" title="c7">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> )
 ;<br/><span class="kw">thus </span><a NAME="E4:73_1_1_1_1_2_2_2"/>
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_1_1_1_2_2_2"><span class="lab"><font color="Green" title="E48">E1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">hence </span><a NAME="E4:73_1_1_1_1_2_2"/>
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> </span> } </span>  is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E46">D</font></span></a>, <a class="txt" href="#E2:73_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E47">E</font></span></a>, <a class="ref" href="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">reconsider </span><font color="Maroon" title="c7">M</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> </span> } </span>  as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c1">F</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1_1_1_1_2"><span class="lab"><font color="Green" title="E45">I</font></span></a></span>;<br/><a NAME="E5:73_1_1_1_1_2"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="73_1_1_1_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c8">m</font> = <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ex <font color="Olive" title="b2">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b2">y</font> );<br/>
<a NAME="E1:73_1_1_1_1_2_3"/><span class="lab"><font color="Green" title="E46">B1</font></span>: 
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y1</font>, <font color="Olive" title="b3">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</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"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">y2</font>] holds <br/><font color="Olive" title="b2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y2</font>
 
;<br/>
<div><span class="lab"><font color="Green" title="E47">B2</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="73_1_1_1_1_2_3_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</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"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/> ex <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</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"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> implies  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/><span class="kw">assume </span><a NAME="E1:73_1_1_1_1_2_3_1"/>
<font color="Maroon" title="c9">x</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"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">i</font> = <font color="Maroon" title="c9">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ;<br/><a NAME="E3:73_1_1_1_1_2_3_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">z</font> = <font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a></span>;<br/><span class="kw">thus </span><a NAME="E5:73_1_1_1_1_2_3_1"/>
 ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="73_1_1_1_1_2_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
<font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c11">z</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c11">z</font>]</span><br/>

<span class="kw">thus </span><a NAME="E1:73_1_1_1_1_2_3_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c11">z</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
<span class="kw">consider </span><font color="Maroon" title="c9">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E4:73_1_1_1_1_2_3"/><span class="lab"><font color="Green" title="E48">C</font></span>: 
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c9">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"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; (  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</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"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_1.html#S2" onmouseover="rs('funct_1/S2')" onmouseout="rh()">FUNCT_1:sch 2</a>(<a class="txt" href="#E1:73_1_1_1_1_2_3"><span class="lab"><font color="Green" title="E46">B1</font></span></a>, <a class="txt" href="#E2:73_1_1_1_1_2_3"><span class="lab"><font color="Green" title="E47">B2</font></span></a>);<br/></span>
<div><span class="lab"><font color="Green" title="E49">A1</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="73_1_1_1_1_2_3_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font> holds <br/><font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c9">f</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font> implies <font color="Maroon" title="c10">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c9">f</font> )</span><br/><span class="kw">assume </span><a NAME="E1:73_1_1_1_1_2_3_2"/>
<font color="Maroon" title="c10">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c9">f</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:73_1_1_1_1_2_3_2"/><span class="lab"><font color="Green" title="E50">A2</font></span>: 
( <font color="Maroon" title="c10">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> )
 ;<br/><a NAME="E4:73_1_1_1_1_2_3_2"/>
( <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_1_1_1_2_3_2"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">x</font> = <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/><a NAME="E6:73_1_1_1_1_2_3_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_1_1_1_1_2_3"><span class="lab"><font color="Green" title="E48">C</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:73_1_1_1_1_2_3_2"/>
<font color="Maroon" title="c10">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c9">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_1_1_1_1_2_3"><span class="lab"><font color="Green" title="E48">C</font></span></a>, <a class="txt" href="#E3:73_1_1_1_1_2_3_2"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="73_1_1_1_1_2_3_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c9">f</font> holds <br/><font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c9">f</font> implies <font color="Maroon" title="c10">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font> )</span><br/><span class="kw">assume </span><a NAME="E1:73_1_1_1_1_2_3_3"/>
<font color="Maroon" title="c10">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c9">f</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:73_1_1_1_1_2_3_3"/><span class="lab"><font color="Green" title="E50">A2</font></span>: 
( <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c9">f</font> &amp; <font color="Maroon" title="c10">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">u</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/><a NAME="E4:73_1_1_1_1_2_3_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">u</font>,<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">u</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_1_1_1_1_2_3"><span class="lab"><font color="Green" title="E48">C</font></span></a>, <a class="txt" href="#E3:73_1_1_1_1_2_3_3"><span class="lab"><font color="Green" title="E50">A2</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>, <font color="Maroon" title="c13">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E6:73_1_1_1_1_2_3_3"/><span class="lab"><font color="Green" title="E51">A3</font></span>: 
( <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c13">y</font> )
 ;<br/><a NAME="E7:73_1_1_1_1_2_3_3"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">x</font> &amp; <font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/><a NAME="E8:73_1_1_1_1_2_3_3"/><span class="kw">then </span>
<font color="Maroon" title="c13">y</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"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:73_1_1_1_1_2_3_3"><span class="lab"><font color="Green" title="E51">A3</font></span></a>, <a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E9:73_1_1_1_1_2_3_3"/><span class="kw">then </span>
<font color="Maroon" title="c13">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E10:73_1_1_1_1_2_3_3"/>
<font color="Maroon" title="c10">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_1_1_1_2_3_3"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="txt" href="#E6:73_1_1_1_1_2_3_3"><span class="lab"><font color="Green" title="E51">A3</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E7:73_1_1_1_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_1_1_1_1_2_3"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="ref" href="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI:2</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="73_1_1_1_1_2_3_4"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:73_1_1_1_1_2_3_4"/>
 not <font color="Maroon" title="c9">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">x1</font>, <font color="Maroon" title="c11">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:73_1_1_1_1_2_3_4"/><span class="lab"><font color="Green" title="E51">A1</font></span>: 
( <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c9">f</font> &amp; <font color="Maroon" title="c11">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c9">f</font> &amp; <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font> &amp; <font color="Maroon" title="c10">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">x2</font> )
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c12">n1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>, <font color="Maroon" title="c13">y1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E5:73_1_1_1_1_2_3_4"/><span class="lab"><font color="Green" title="E52">A2</font></span>: 
( <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">n1</font> &amp; <font color="Maroon" title="c13">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">n1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c13">y1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_1_1_1_2_3_4"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="txt" href="#E4:73_1_1_1_1_2_3"><span class="lab"><font color="Green" title="E48">C</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c14">n2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>, <font color="Maroon" title="c15">y2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E7:73_1_1_1_1_2_3_4"/><span class="lab"><font color="Green" title="E53">A3</font></span>: 
( <font color="Maroon" title="c11">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">n2</font> &amp; <font color="Maroon" title="c15">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">n2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c15">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_1_1_1_2_3_4"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="txt" href="#E4:73_1_1_1_1_2_3"><span class="lab"><font color="Green" title="E48">C</font></span></a></span>;<br/><a NAME="E8:73_1_1_1_1_2_3_4"/>
( <font color="Maroon" title="c12">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c14">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/><a NAME="E9:73_1_1_1_1_2_3_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A4</font></span>: 
( <font color="Maroon" title="c13">y1</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"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <font color="Maroon" title="c15">y2</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"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:73_1_1_1_1_2_3_4"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E5:73_1_1_1_1_2_3_4"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E10:73_1_1_1_1_2_3_4"/><span class="lab"><font color="Green" title="E55">A5</font></span>: 
<font color="Maroon" title="c13">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">y2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_1_1_1_2_3_4"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="txt" href="#E5:73_1_1_1_1_2_3_4"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E7:73_1_1_1_1_2_3_4"><span class="lab"><font color="Green" title="E53">A3</font></span></a></span>;<br/><a NAME="E11:73_1_1_1_1_2_3_4"/><span class="lab"><font color="Green" title="E56">A6</font></span>: 
( <font color="Maroon" title="c13">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c15">y2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:73_1_1_1_1_2_3_4"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1_1_1_2_3_4_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:73_1_1_1_1_2_3_4_1"/>
( <font color="Maroon" title="c13">y1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">y2</font> or <font color="Maroon" title="c13">y1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c15">y2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:73_1_1_1_1_2_3_4"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1_1_1_2_3_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:73_1_1_1_1_2_3_4_1_1"/>
<font color="Maroon" title="c13">y1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">y2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:73_1_1_1_1_2_3_4_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_1_1_1_1_2"><span class="lab"><font color="Green" title="E44">U</font></span></a>, <a class="txt" href="#E3:73_1_1_1_1_2_3_4"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="txt" href="#E5:73_1_1_1_1_2_3_4"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E7:73_1_1_1_1_2_3_4"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E11:73_1_1_1_1_2_3_4"><span class="lab"><font color="Green" title="E56">A6</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="73_1_1_1_1_2_3_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:73_1_1_1_1_2_3_4_1_2"/>
<font color="Maroon" title="c13">y1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c15">y2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:73_1_1_1_1_2_3_4_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_1_1_1_1_2"><span class="lab"><font color="Green" title="E44">U</font></span></a>, <a class="txt" href="#E3:73_1_1_1_1_2_3_4"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="txt" href="#E5:73_1_1_1_1_2_3_4"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E7:73_1_1_1_1_2_3_4"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E11:73_1_1_1_1_2_3_4"><span class="lab"><font color="Green" title="E56">A6</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">then </span> <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">M</font> = 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:73_1_1_1_1_2_3"><span class="lab"><font color="Green" title="E50">A</font></span></a>, <a class="txt" href="#E4:73_1_1_1_1_2_3"><span class="lab"><font color="Green" title="E48">C</font></span></a>, <a class="ref" href="card_1.html#T70" onmouseover="rs('card_1/T70')" onmouseout="rh()">CARD_1:70</a></span>
<br/>.= 
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T57" onmouseover="rs('finseq_1/T57')" onmouseout="rh()">FINSEQ_1:57</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E10:73_1_1_1_1_2_3"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E6:73_1_1_1_1_2"/><span class="kw">then </span>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">M</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E7:73_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">M</font> is  <a href="vectsp_7.html#NV2" title="VECTSP_7:NV.2">linearly-dependent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_1"><span class="lab"><font color="Green" title="E43">H</font></span></a>, <a class="txt" href="#E1:73_1_1"><span class="lab"><font color="Green" title="E42">AS</font></span></a>, <a class="txt" href="field_7.html#E50"><span class="lab"><font color="Green" title="E41">lemlin</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">l</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c7">M</font><span class="kw"> such that </span><br/><a NAME="E9:73_1_1_1_1_2"/><span class="lab"><font color="Green" title="E46">D1</font></span>: 
(  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c8">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c1">F</font>)</span>)</span> &amp;  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c8">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_7.html#D1" onmouseover="rs('vectsp_7/D1')" onmouseout="rh()">VECTSP_7:def 1</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c9">z</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c8">l</font>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c1">F</font>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E11:73_1_1_1_1_2"/><span class="lab"><font color="Green" title="E47">D2</font></span>: 
(  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c8">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v</font> &amp; <font color="Maroon" title="c8">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:73_1_1_1_1_2"><span class="lab"><font color="Green" title="E46">D1</font></span></a>, <a class="ref" href="vectsp_6.html#T1" onmouseover="rs('vectsp_6/T1')" onmouseout="rh()">VECTSP_6:1</a></span>;<br/><a NAME="E12:73_1_1_1_1_2"/>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c8">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_6.html#D4" onmouseover="rs('vectsp_6/D4')" onmouseout="rh()">VECTSP_6:def 4</a></span>;<br/><a NAME="E13:73_1_1_1_1_2"/><span class="kw">then </span>
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c8">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:73_1_1_1_1_2"><span class="lab"><font color="Green" title="E46">D1</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E15:73_1_1_1_1_2"/><span class="lab"><font color="Green" title="E48">D3</font></span>: 
(  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c8">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> )
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c12">pM</font> being   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">F</font><span class="kw"> such that </span><br/><a NAME="E17:73_1_1_1_1_2"/><span class="lab"><font color="Green" title="E49">D4a</font></span>: 
(  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c12">pM</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> holds <br/><font color="Maroon" title="c12">pM</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b1">i</font></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_6.html#T47" onmouseover="rs('field_6/T47')" onmouseout="rh()">FIELD_6:47</a></span>;<br/><a NAME="E18:73_1_1_1_1_2"/>
<font color="Maroon" title="c12">pM</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:73_1_1_1_1_2"><span class="lab"><font color="Green" title="E48">D3</font></span></a>, <a class="txt" href="#E11:73_1_1_1_1_2"><span class="lab"><font color="Green" title="E47">D2</font></span></a>, <a class="txt" href="#E17:73_1_1_1_1_2"><span class="lab"><font color="Green" title="E49">D4a</font></span></a></span>;<br/><a NAME="E19:73_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">pM</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <font color="Maroon" title="c1">F</font>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">pM</font> = <font color="Maroon" title="c12">pM</font> as   non  <a href="ratfunc1.html#NV1" title="RATFUNC1:NV.1">zero</a>  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">F</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#D5" onmouseover="rs('uproots/D5')" onmouseout="rh()">UPROOTS:def 5</a></span>;<br/> <a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<font color="Maroon" title="c13">pM</font>,<font color="Maroon" title="c4">a</font>) = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c1">F</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_1_1_1_1_2"><span class="lab"><font color="Green" title="E44">U</font></span></a>, <a class="txt" href="#E9:73_1_1_1_1_2"><span class="lab"><font color="Green" title="E46">D1</font></span></a>, <a class="txt" href="#E17:73_1_1_1_1_2"><span class="lab"><font color="Green" title="E49">D4a</font></span></a>, <a class="ref" href="field_6.html#T49" onmouseover="rs('field_6/T49')" onmouseout="rh()">FIELD_6:49</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">E</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#D6" onmouseover="rs('field_4/D6')" onmouseout="rh()">FIELD_4:def 6</a></span>
;<br/><span class="kw">hence </span><a NAME="E22:73_1_1_1_1_2"/>
<font color="Maroon" title="c4">a</font> is <font color="Maroon" title="c1">F</font> <a href="field_6.html#V2" title="FIELD_6:attr.2">-algebraic</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_6.html#T43" onmouseover="rs('field_6/T43')" onmouseout="rh()">FIELD_6:43</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E5:73_1_1"/>
<font color="Maroon" title="c2">E</font> is <font color="Maroon" title="c1">F</font> <a href="field_7.html#V2" title="FIELD_7:attr.2">-algebraic</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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