<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">V</font>, <font color="Maroon" title="c2">A</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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">m</font>, <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> holds <br/><span class="p1">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span><br/><span class="kw">let </span><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="c3">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> implies <span class="p1">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:23"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
<font color="Maroon" title="c3">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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"> (  not <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> or <span class="p1">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">S</font> =  <a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</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="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 implies <span class="p1">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 );<br/>
<a NAME="E2:23"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
;<br/>
<a NAME="E3:23"/><span class="lab"><font color="Green" title="E14">A3</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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="23_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">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">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:23_1"/><span class="lab"><font color="Green" title="E15">A4</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</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="c6">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]</span><br/>

<span class="kw">assume </span><a NAME="E2:23_1"/><span class="lab"><font color="Green" title="E16">A5</font></span>: 
<font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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"> <span class="p1">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_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="E1:23_1_1"/>
( <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 or <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="ref" href="nat_1.html#T8" onmouseover="rs('nat_1/T8')" onmouseout="rh()">NAT_1:8</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:23_1_1_1"/>
<font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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"> <span class="p1">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:23_1_1_1"/>
<span class="p1">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:23_1_1_2"/><span class="lab"><font color="Green" title="E17">A6</font></span>: 
<font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</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"> <span class="p1">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_2_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="E1:23_1_1_2_1"/>
( <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c6">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:23_1_1_2_1_1"/>
<font color="Maroon" title="c6">k</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"> <span class="p1">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:23_1_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E1:23_1_1_2"><span class="lab"><font color="Green" title="E17">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="23_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:23_1_1_2_1_2"/><span class="lab"><font color="Green" title="E18">A7</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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"> <span class="p1">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span><br/><div class="add"><span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span>( $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies <span class="p1">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> );<br/><a NAME="E2:23_1_1_2_1_2"/><span class="lab"><font color="Green" title="E19">A8</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 ;<br/><a NAME="E3:23_1_1_2_1_2"/><span class="lab"><font color="Green" title="E20">A9</font></span>: 
 for <font color="Olive" title="b1">z</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">z</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">z</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">z</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">S<sub>2</sub></font>[<font color="Maroon" title="c7">z</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">z</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:23_1_1_2_1_2_1"/><span class="lab"><font color="Green" title="E21">A10</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">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>2</sub></font>[<font color="Maroon" title="c7">z</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_2_1_2_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="E1:23_1_1_2_1_2_1_1"/>
( <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c7">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_2_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:23_1_1_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E22">A11</font></span>: 
<font color="Maroon" title="c7">z</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="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">z</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:23_1_1_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E23">A12</font></span>: 
<span class="p1">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nomin_1.html#K1" title="NOMIN_1:func.1">NDSS</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nomin_1.html#T9" target="_self" onmouseover="rs('nomin_1/T9')" onmouseout="rh()">Th9</a></span>;<br/><a NAME="E3:23_1_1_2_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nomin_1.html#K1" title="NOMIN_1:func.1">NDSS</a> (<font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="nomin_1.html#K1" title="NOMIN_1:func.1">NDSS</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nomin_1.html#T10" target="_self" onmouseover="rs('nomin_1/T10')" onmouseout="rh()">Th10</a></span>;<br/><span class="kw">hence </span><a NAME="E4:23_1_1_2_1_2_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">z</font> <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="#E1:23_1_1_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="txt" href="#E2:23_1_1_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E23">A12</font></span></a>, <a class="ref" href="nomin_1.html#T7" target="_self" onmouseover="rs('nomin_1/T7')" onmouseout="rh()">Th7</a>, <a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</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="23_1_1_2_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:23_1_1_2_1_2_1_1_2"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">z</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:23_1_1_2_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A13</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">z</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E21">A10</font></span></a>, <a class="ref" href="xboole_1.html#T9" onmouseover="rs('xboole_1/T9')" onmouseout="rh()">XBOOLE_1:9</a></span>;<br/><a NAME="E3:23_1_1_2_1_2_1_1_2"/><span class="lab"><font color="Green" title="E23">A14</font></span>: 
<span class="p1">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">z</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nomin_1.html#K1" title="NOMIN_1:func.1">NDSS</a> (<font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nomin_1.html#D3" target="_self" onmouseover="rs('nomin_1/D3')" onmouseout="rh()">Def3</a></span>;<br/><a NAME="E4:23_1_1_2_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">z</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nomin_1.html#K1" title="NOMIN_1:func.1">NDSS</a> (<font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">z</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nomin_1.html#D3" target="_self" onmouseover="rs('nomin_1/D3')" onmouseout="rh()">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E5:23_1_1_2_1_2_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">z</font> <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="#E2:23_1_1_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E22">A13</font></span></a>, <a class="txt" href="#E3:23_1_1_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E23">A14</font></span></a>, <a class="ref" href="nomin_1.html#T7" target="_self" onmouseover="rs('nomin_1/T7')" onmouseout="rh()">Th7</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><a NAME="E4:23_1_1_2_1_2"/>
 for <font color="Olive" title="b1">z</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">z</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="#E2:23_1_1_2_1_2"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="txt" href="#E3:23_1_1_2_1_2"><span class="lab"><font color="Green" title="E20">A9</font></span></a>);<br/></span><a NAME="E5:23_1_1_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1_2_1_2"><span class="lab"><font color="Green" title="E18">A7</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:23_1_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E1:23_1_1_2"><span class="lab"><font color="Green" title="E17">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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E4:23"/>
 for <font color="Olive" title="b1">k</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">k</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="#E2:23"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E14">A3</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E5:23"/>
(  not <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> or <span class="p1">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="nomin_1.html#K2" title="NOMIN_1:func.2">FNDSC</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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"> verum</span><br/>


</div>
