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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 &amp; <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">n</font>];<br/>
<a NAME="E1:5_1"/><i><font color="Green" title="E6">A3</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st ( for <font color="Olive" title="b2">r</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">r</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">r</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font>] )</span><br/>

<b>assume </b><a NAME="E1:5_1_1"/><i><font color="Green" title="E7">A4</font></i>: 
for <font color="Olive" title="b1">r</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font>]</span><br/>

<b>let </b><font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font>] )</span><br/>



<b>assume </b><a NAME="E2:5_1_1"/><i><font color="Green" title="E8">A5</font></i>: 
( <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font>]</span><br/>

<b>set </b><font color="Maroon" title="c4">s</font> =  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font>;<br/>
<a NAME="E3:5_1_1"/><i><font color="Green" title="E9">A6</font></i>: 
 <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font>
 
<b>by </b><i><a class="txt" href="fib_num.html#E2:5_1_1"><i><font color="Green" title="E8">A5</font></i></a>, <a class="ref" href="square_1.html#T50" title="SQUARE_1:th.50">SQUARE_1:50</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_1_1_1"/>
(  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> or  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> )
 </span><b>by </b><i><a class="txt" href="fib_num.html#E3:5_1_1"><i><font color="Green" title="E9">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_1_1"><b>suppose </b></a><a NAME="E1:5_1_1_1_1"/>
 <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font>]</span><br/><div class="add"><a NAME="E2:5_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font> implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font>] )
 <b>by </b><i><a class="txt" href="fib_num.html#E1:5_1_1"><i><font color="Green" title="E7">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E3:5_1_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font>]
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_1_2"><b>suppose </b></a><a NAME="E1:5_1_1_1_2"/><i><font color="Green" title="E10">A7</font></i>: 
 <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font>]</span><br/><div class="add"><a NAME="E2:5_1_1_1_2"/><i><font color="Green" title="E11">A8</font></i>: 
for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">n</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">m</font>, <font color="Maroon" title="c6">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">n</font>] )</span><br/>



<b>assume </b><a NAME="E1:5_1_1_1_2_1"/><i><font color="Green" title="E12">A9</font></i>: 
( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">n</font>]</span><br/>

<b>set </b><font color="Maroon" title="c7">s</font> =  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">n</font>;<br/>
<a NAME="E2:5_1_1_1_2_1"/><i><font color="Green" title="E13">A10</font></i>: 
 <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font>
 
<b>by </b><i><a class="txt" href="fib_num.html#E1:5_1_1_1_2_1"><i><font color="Green" title="E12">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T16" title="XXREAL_0:th.16">XXREAL_0:16</a></i>;<br/>
<a NAME="E3:5_1_1_1_2_1"/><i><font color="Green" title="E14">A11</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">n</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/>
<a NAME="E4:5_1_1_1_2_1"/>
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">n</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/>
<b>hence </b><a NAME="E5:5_1_1_1_2_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">n</font>]
 <b>by </b><i><a class="txt" href="fib_num.html#E1:5_1_1"><i><font color="Green" title="E7">A4</font></i></a>, <a class="txt" href="fib_num.html#E2:5_1_1_1_2_1"><i><font color="Green" title="E13">A10</font></i></a>, <a class="txt" href="fib_num.html#E3:5_1_1_1_2_1"><i><font color="Green" title="E14">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:5_1_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_1_1_1_2_2_1"/>
( <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> or <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> )
 </span><b>by </b><i><a class="txt" href="fib_num.html#E1:5_1_1_1_2"><i><font color="Green" title="E10">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T16" title="XXREAL_0:th.16">XXREAL_0:16</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:5_1_1_1_2_2_1_1"/>
<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font>]</span><br/><div class="add"><b>hence </b><a NAME="E2:5_1_1_1_2_2_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font>]
 <b>by </b><i><a class="txt" href="fib_num.html#E2:5"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="fib_num.html#E2:5_1_1"><i><font color="Green" title="E8">A5</font></i></a>, <a class="txt" href="fib_num.html#E2:5_1_1_1_2"><i><font color="Green" title="E11">A8</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:5_1_1_1_2_2_1_2"/>
<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font>]</span><br/><div class="add"><a NAME="E2:5_1_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">m</font>]
 <b>by </b><i><a class="txt" href="fib_num.html#E2:5"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="fib_num.html#E2:5_1_1"><i><font color="Green" title="E8">A5</font></i></a>, <a class="txt" href="fib_num.html#E2:5_1_1_1_2"><i><font color="Green" title="E11">A8</font></i></a></i>;<br/><b>hence </b><a NAME="E3:5_1_1_1_2_2_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font>]
 <b>by </b><i><a class="txt" href="fib_num.html#E1:5"><i><font color="Green" title="E4">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:5_1"/><i><font color="Green" title="E7">A12</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S4" title="NAT_1:sch.4">NAT_1:sch 4</a>(<a class="txt" href="fib_num.html#E1:5_1"><i><font color="Green" title="E6">A3</font></i></a>);<br/></i>
<b>let </b><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">n</font>]</span><br/>



<b>set </b><font color="Maroon" title="c3">k</font> =  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">n</font>;<br/>
<a NAME="E3:5_1"/>
( <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">n</font> implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">n</font>] )
 
<b>by </b><i><a class="txt" href="fib_num.html#E2:5_1"><i><font color="Green" title="E7">A12</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:5_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">n</font>]
 <b>by </b><i><a class="ref" href="square_1.html#T50" title="SQUARE_1:th.50">SQUARE_1:50</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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