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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="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="E1:22_1"/><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">n</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies <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>assume </b><a NAME="E1:22_1_1"/>
<font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</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/>

<a NAME="E2:22_1_1"/><b>then </b>
( <font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 
<b>by </b><i><a class="ref" href="nat_1.html#T7" title="NAT_1:th.7">NAT_1:7</a></i>;<br/>
<b>hence </b><a NAME="E3:22_1_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="txt" href="ramsey_1.html#E1:22"><i><font color="Green" title="E18">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>
<a NAME="E2:22_1"/><i><font color="Green" title="E21">A4</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="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="real_1.html#K7" title="REAL_1:func.7">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><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>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:22_1_2"/><i><font color="Green" title="E22">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<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">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1]</span><br/>

<a NAME="E2:22_1_2"/>
for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</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>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">n</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <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 href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 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="E1:22_1_2_1"/><i><font color="Green" title="E23">A6</font></i>: 
<font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <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 href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 ; <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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_2_1_1"/>
( <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or ( <font color="Maroon" title="c2">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:22_1_2_1_1_1"/>
( <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</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="c2">m</font>,<font color="Maroon" title="c3">n</font>]</span><br/><div class="add"><b>hence </b><a NAME="E2:22_1_2_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="txt" href="ramsey_1.html#E1:22"><i><font color="Green" title="E18">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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:22_1_2_1_1_2"/><i><font color="Green" title="E24">A7</font></i>: 
( <font color="Maroon" title="c2">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</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="c2">m</font>,<font color="Maroon" title="c3">n</font>]</span><br/><div class="add"><a NAME="E2:22_1_2_1_1_2"/><b>then </b>
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">n</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c4">n'</font> = <font color="Maroon" title="c3">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="nat_1.html#T20" title="NAT_1:th.20">NAT_1:20</a></i>;<br/><a NAME="E4:22_1_2_1_1_2"/><i><font color="Green" title="E25">A8</font></i>: 
<font color="Maroon" title="c2">m</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">n'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E1:22_1_2_1"><i><font color="Green" title="E23">A6</font></i></a></i>;<br/><a NAME="E5:22_1_2_1_1_2"/>
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">m</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E1:22_1_2_1_1_2"><i><font color="Green" title="E24">A7</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c5">m'</font> = <font color="Maroon" title="c2">m</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="nat_1.html#T20" title="NAT_1:th.20">NAT_1:20</a></i>;<br/><a NAME="E7:22_1_2_1_1_2"/>
<font color="Maroon" title="c5">m'</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E1:22_1_2_1"><i><font color="Green" title="E23">A6</font></i></a></i>;<br/><a NAME="E8:22_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">m'</font>,<font color="Maroon" title="c4">n'</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">m'</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1,<font color="Maroon" title="c4">n'</font>] )
 <b>by </b><i><a class="txt" href="ramsey_1.html#E1:22_1_2"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="ramsey_1.html#E4:22_1_2_1_1_2"><i><font color="Green" title="E25">A8</font></i></a></i>;<br/><b>hence </b><a NAME="E9:22_1_2_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>]
 <b>by </b><i><a class="txt" href="ramsey_1.html#E2:22"><i><font color="Green" title="E19">A2</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>
<b>hence </b><a NAME="E3:22_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1]
 ; <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>
<a NAME="E3:22_1"/><i><font color="Green" title="E22">A9</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  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#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="ramsey_1.html#E1:22_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="ramsey_1.html#E2:22_1"><i><font color="Green" title="E21">A4</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="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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> = <font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">n</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">k</font> = <font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E5:22_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>]
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E3:22_1"><i><font color="Green" title="E22">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:22_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
