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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="bcialg_1.html#NM1" title="BCIALG_1:NM.1">BCI-algebra</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">  for <font color="Olive" title="b1">j</font>, <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c1">X</font> is   <a href="bcialg_1.html#V19" title="BCIALG_1:attr.19">p-Semisimple</a>  <a href="bcialg_1.html#NM1" title="BCIALG_1:NM.1">BCI-algebra</a> holds <br/><font color="Maroon" title="c1">X</font> is    <a href="bcialg_5.html#M1" title="BCIALG_5:mode.1">BCI-algebra</a> of <font color="Olive" title="b3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">j</font>,<font color="Olive" title="b3">n</font>,<font color="Olive" title="b2">m</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span><br/><span class="kw">let </span><font color="Maroon" title="c2">j</font>, <font color="Maroon" title="c3">m</font>, <font color="Maroon" title="c4">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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="c1">X</font> is   <a href="bcialg_1.html#V19" title="BCIALG_1:attr.19">p-Semisimple</a>  <a href="bcialg_1.html#NM1" title="BCIALG_1:NM.1">BCI-algebra</a> implies <font color="Maroon" title="c1">X</font> is    <a href="bcialg_5.html#M1" title="BCIALG_5:mode.1">BCI-algebra</a> of <font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c3">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )</span><br/>







<span class="kw">assume </span><a NAME="E1:51"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
<font color="Maroon" title="c1">X</font> is   <a href="bcialg_1.html#V19" title="BCIALG_1:attr.19">p-Semisimple</a>  <a href="bcialg_1.html#NM1" title="BCIALG_1:NM.1">BCI-algebra</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="c1">X</font> is    <a href="bcialg_5.html#M1" title="BCIALG_5:mode.1">BCI-algebra</a> of <font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c3">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span><br/>

<a NAME="E2:51"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> holds   <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c4">n</font>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</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">  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font></span><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>( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> implies  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> ($1,$1,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> );<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_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">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st ( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> implies  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<font color="Olive" title="b1">k</font>,<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> ) &amp; <font color="Olive" title="b1">k</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="c4">n</font> holds <br/> <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> implies  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> ) &amp; <font color="Maroon" title="c7">k</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="c4">n</font> implies  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> )</span><br/><span class="kw">assume </span><a NAME="E1:51_1_1"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
( <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> implies  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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"> ( <font color="Maroon" title="c7">k</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="c4">n</font> implies  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> )</span><br/><span class="kw">set </span><font color="Maroon" title="c8">m</font> = <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/><span class="lab"><font color="Green" title="E29">A4</font></span>:  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) = 
<span class="p1">(<span class="default"><a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<font color="Maroon" title="c7">k</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>)</span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c6">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_5.html#T9" target="_self" onmouseover="rs('bcialg_5/T9')" onmouseout="rh()">Th9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>)</span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c6">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_5.html#T10" target="_self" onmouseover="rs('bcialg_5/T10')" onmouseout="rh()">Th10</a></span>
;<br/><span class="kw">assume </span><a NAME="E3:51_1_1"/>
<font color="Maroon" title="c7">k</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="c4">n</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">  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font></span><br/><a NAME="E4:51_1_1"/><span class="kw">then </span>
 <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c6">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E1:51_1_1"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E2:51_1_1"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="ref" href="bcialg_1.html#D26" onmouseover="rs('bcialg_1/D26')" onmouseout="rh()">BCIALG_1:def 26</a>, <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="E5:51_1_1"/>
 <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="bcialg_1.html#D26" onmouseover="rs('bcialg_1/D26')" onmouseout="rh()">BCIALG_1:def 26</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="E2:51_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A5</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
;<br/>
 <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) = 
<font color="Maroon" title="c5">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c6">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_5.html#T7" target="_self" onmouseover="rs('bcialg_5/T7')" onmouseout="rh()">Th7</a></span>
<br/>.= 
<font color="Maroon" title="c6">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="bcialg_1.html#D26" onmouseover="rs('bcialg_1/D26')" onmouseout="rh()">BCIALG_1:def 26</a></span>
;<br/>
<a NAME="E4:51_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A6</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
;<br/>
<a NAME="E5:51_1"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E4:51_1"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E28">A5</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E6:51_1"/>
 <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:51"/><span class="lab"><font color="Green" title="E28">A7</font></span>: 
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> holds   <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<font color="Maroon" title="c3">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</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">  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<font color="Maroon" title="c3">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font></span><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>( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> implies  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> ($1,<span class="p1">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> );<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_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">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st ( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> implies  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> ) &amp; <font color="Olive" title="b1">k</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">m</font> holds <br/> <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> implies  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<font color="Maroon" title="c7">k</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> ) &amp; <font color="Maroon" title="c7">k</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">m</font> implies  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:51_2_1"/><span class="lab"><font color="Green" title="E29">A8</font></span>: 
( <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> implies  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<font color="Maroon" title="c7">k</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</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="c7">k</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">m</font> implies  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> )</span><br/><span class="kw">set </span><font color="Maroon" title="c8">l</font> = <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/><span class="lab"><font color="Green" title="E30">A9</font></span>:  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) = 
<span class="p1">(<span class="default"><a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<font color="Maroon" title="c7">k</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>)</span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c6">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_5.html#T9" target="_self" onmouseover="rs('bcialg_5/T9')" onmouseout="rh()">Th9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<font color="Maroon" title="c7">k</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>)</span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c6">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_5.html#T10" target="_self" onmouseover="rs('bcialg_5/T10')" onmouseout="rh()">Th10</a></span>
;<br/><span class="kw">assume </span><a NAME="E3:51_2_1"/>
<font color="Maroon" title="c7">k</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">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">  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font></span><br/><span class="kw">then </span> <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E2:51_2_1"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="ref" href="bcialg_1.html#T7" onmouseover="rs('bcialg_1/T7')" onmouseout="rh()">BCIALG_1:7</a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c6">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="bcialg_1.html#D26" onmouseover="rs('bcialg_1/D26')" onmouseout="rh()">BCIALG_1:def 26</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:51_2_1"/>
 <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="bcialg_1.html#D26" onmouseover="rs('bcialg_1/D26')" onmouseout="rh()">BCIALG_1:def 26</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="E2:51_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A10</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
;<br/>
 <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) = 
<span class="p1">(<span class="default"><a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>)</span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_5.html#T10" target="_self" onmouseover="rs('bcialg_5/T10')" onmouseout="rh()">Th10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_5.html#T7" target="_self" onmouseover="rs('bcialg_5/T7')" onmouseout="rh()">Th7</a></span>
<br/>.= 
<font color="Maroon" title="c6">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="bcialg_1.html#D26" onmouseover="rs('bcialg_1/D26')" onmouseout="rh()">BCIALG_1:def 26</a></span>
<br/>.= 
<font color="Maroon" title="c5">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="bcialg_1.html#D26" onmouseover="rs('bcialg_1/D26')" onmouseout="rh()">BCIALG_1:def 26</a></span>
;<br/>
<a NAME="E4:51_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A11</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
;<br/>
<a NAME="E5:51_2"/>
 for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E4:51_2"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="txt" href="#E2:51_2"><span class="lab"><font color="Green" title="E29">A10</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E6:51_2"/>
 <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<font color="Maroon" title="c3">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</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>
<a NAME="E4:51"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> holds   <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span>,<font color="Maroon" title="c4">n</font>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<font color="Maroon" title="c3">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</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">  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span>,<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<font color="Maroon" title="c3">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font>)</span><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>( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">j</font> implies  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> $1</span>)</span>,<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<font color="Maroon" title="c3">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> $1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font>) );<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_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">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st ( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">j</font> implies  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">k</font></span>)</span>,<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<font color="Maroon" title="c3">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font>) ) &amp; <font color="Olive" title="b1">k</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="c2">j</font> holds <br/> <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<font color="Maroon" title="c3">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font>)</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">j</font> implies  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span>,<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<font color="Maroon" title="c3">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font>) ) &amp; <font color="Maroon" title="c7">k</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="c2">j</font> implies  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<font color="Maroon" title="c3">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font>) )</span><br/><span class="kw">assume </span><a NAME="E1:51_3_1"/><span class="lab"><font color="Green" title="E29">A12</font></span>: 
( <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">j</font> implies  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span>,<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<font color="Maroon" title="c3">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</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="c7">k</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="c2">j</font> implies  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<font color="Maroon" title="c3">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font>) )</span><br/><span class="kw">set </span><font color="Maroon" title="c8">l</font> = <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/><span class="kw">assume </span><a NAME="E2:51_3_1"/>
<font color="Maroon" title="c7">k</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="c2">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">  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<font color="Maroon" title="c3">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font>)</span><br/><span class="kw">then </span> <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) = 
<span class="p1">(<span class="default"><a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<font color="Maroon" title="c3">m</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c6">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_3_1"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="ref" href="bcialg_5.html#T9" target="_self" onmouseover="rs('bcialg_5/T9')" onmouseout="rh()">Th9</a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>
<br/>.= 
 <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<font color="Maroon" title="c3">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_5.html#T10" target="_self" onmouseover="rs('bcialg_5/T10')" onmouseout="rh()">Th10</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:51_3_1"/>
 <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<font color="Maroon" title="c3">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</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>
<a NAME="E2:51_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A13</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
;<br/>
 <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) = 
<font color="Maroon" title="c6">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51"><span class="lab"><font color="Green" title="E27">A2</font></span></a></span>
<br/>.= 
 <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<font color="Maroon" title="c3">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E28">A7</font></span></a></span>
;<br/>
<a NAME="E4:51_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A14</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
;<br/>
<a NAME="E5:51_3"/>
 for <font color="Olive" title="b1">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E4:51_3"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="txt" href="#E2:51_3"><span class="lab"><font color="Green" title="E29">A13</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E6:51_3"/>
 <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span>,<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<font color="Maroon" title="c3">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</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>
<span class="kw">hence </span><a NAME="E5:51"/>
<font color="Maroon" title="c1">X</font> is    <a href="bcialg_5.html#M1" title="BCIALG_5:mode.1">BCI-algebra</a> of <font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c3">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span> <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="bcialg_5.html#D3" target="_self" onmouseover="rs('bcialg_5/D3')" onmouseout="rh()">Def3</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>
