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<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">k</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"> not <!--HUMANRECFAILED-->( ( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">k</font> &amp; ( for <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font>  holds <br/>(  not <font color="Olive" title="b1">f</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> or  not <font color="Olive" title="b1">f</font> is <a href="stirl2_1.html#V1" title="STIRL2_1:attr.1">"increasing</a> ) ) )</span><br/>



<b>assume </b><a NAME="E1:28"/><i><font color="Green" title="E17">A1</font></i>: 
( ( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">k</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font> st <br/>( <font color="Olive" title="b1">f</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> &amp; <font color="Olive" title="b1">f</font> is <a href="stirl2_1.html#V1" title="STIRL2_1:attr.1">"increasing</a> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1_1"/>
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1"><b>suppose </b></a><a NAME="E1:28_1_1_1"/><i><font color="Green" title="E18">A2</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font> st <br/>( <font color="Olive" title="b1">f</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> &amp; <font color="Olive" title="b1">f</font> is <a href="stirl2_1.html#V1" title="STIRL2_1:attr.1">"increasing</a> )</span><br/><div class="add"><b>set </b><font color="Maroon" title="c3">f</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/><a NAME="E2:28_1_1_1"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:28"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="stirl2_1.html#E1:28_1_1_1"><i><font color="Green" title="E18">A2</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c4">f</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font> <b>by </b><i><a class="ref" href="funct_2.html#T3" title="FUNCT_2:th.3">FUNCT_2:3</a></i>;<br/><a NAME="E4:28_1_1_1"/>
( <font color="Maroon" title="c4">f</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> &amp; <font color="Maroon" title="c4">f</font> is <a href="stirl2_1.html#V1" title="STIRL2_1:attr.1">"increasing</a> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:28"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="stirl2_1.html#E1:28_1_1_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="stirl2_1.html#T15" target="_self" title="STIRL2_1:th.15">Th15</a></i>;<br/><b>hence </b><a NAME="E5:28_1_1_1"/>
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font> st <br/>( <font color="Olive" title="b1">f</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> &amp; <font color="Olive" title="b1">f</font> is <a href="stirl2_1.html#V1" title="STIRL2_1:attr.1">"increasing</a> )
 ; <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="28_1_1_2"><b>suppose </b></a><a NAME="E1:28_1_1_2"/><i><font color="Green" title="E18">A3</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font> st <br/>( <font color="Olive" title="b1">f</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> &amp; <font color="Olive" title="b1">f</font> is <a href="stirl2_1.html#V1" title="STIRL2_1:attr.1">"increasing</a> )</span><br/><div class="add"><a NAME="E2:28_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:28"><i><font color="Green" title="E17">A1</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c3">k1</font> = <font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#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/><b>reconsider </b><font color="Maroon" title="c4">k</font> = <font color="Maroon" title="c2">k</font>, <font color="Maroon" title="c5">n</font> = <font color="Maroon" title="c1">n</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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>  <b>by </b><i><a class="txt" href="stirl2_1.html#E1:28"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="stirl2_1.html#E1:28_1_1_2"><i><font color="Green" title="E18">A3</font></i></a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">n</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">k</font>] <b>means </b>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> $1,<font color="Maroon" title="c3">k1</font>;<br/><a NAME="E5:28_1_1_2"/><i><font color="Green" title="E19">A4</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">n</font>  ex <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">k</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">k</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<a NAME="E1:28_1_1_2_1"/>
(  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k1</font> &amp; <font color="Maroon" title="c3">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E2:28_1_1_2_1"/><b>then </b>
(  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">k</font> &amp;  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">k1</font> is <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:28_1_1_2_1"/><b>then </b>
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">k1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">k</font>
 
<b>by </b><i><a class="ref" href="euler_1.html#T1" title="EULER_1:th.1">EULER_1:1</a></i>;<br/>
<b>hence </b><a NAME="E4:28_1_1_2_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">k</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Olive" title="b1">y</font>]
 ; <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>consider </b><font color="Maroon" title="c6">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c4">k</font><b> such that </b><br/><a NAME="E7:28_1_1_2"/><i><font color="Green" title="E20">A5</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">n</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>,<font color="Maroon" title="c6">f</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Olive" title="b1">m</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="stirl2_1.html#E5:28_1_1_2"><i><font color="Green" title="E19">A4</font></i></a>);<br/></i><a NAME="E8:28_1_1_2"/><i><font color="Green" title="E21">A6</font></i>: 
<font color="Maroon" title="c6">f</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">m'</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">m'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">k</font> implies <font color="Maroon" title="c7">m'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f</font> )</span><br/><b>assume </b><a NAME="E1:28_1_1_2_2_1"/><i><font color="Green" title="E22">A7</font></i>: 
<font color="Maroon" title="c7">m'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">m'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f</font></span><br/><a NAME="E2:28_1_1_2_2_1"/>
<font color="Maroon" title="c4">k</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="stirl2_1.html#T8" target="_self" title="STIRL2_1:th.8">Th8</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">m</font> = <font color="Maroon" title="c7">m'</font> as    <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>  <b>by </b><i><a class="txt" href="stirl2_1.html#E1:28_1_1_2_2_1"><i><font color="Green" title="E22">A7</font></i></a></i>;<br/><a NAME="E4:28_1_1_2_2_1"/>
<font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:28_1_1_2_2_1"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="euler_1.html#T1" title="EULER_1:th.1">EULER_1:1</a></i>;<br/><a NAME="E5:28_1_1_2_2_1"/><b>then </b>
( <font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k1</font> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:28"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:28_1_1_2_2_1"/><b>then </b>
( <font color="Maroon" title="c8">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">n</font> &amp;  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c8">m</font>,<font color="Maroon" title="c3">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">m</font> )
 <b>by </b><i><a class="ref" href="euler_1.html#T1" title="EULER_1:th.1">EULER_1:1</a>, <a class="ref" href="xxreal_0.html#D8" title="XXREAL_0:def.8">XXREAL_0:def 8</a></i>;<br/><a NAME="E7:28_1_1_2_2_1"/><b>then </b>
( <font color="Maroon" title="c6">f</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">m</font> &amp; <font color="Maroon" title="c8">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E7:28_1_1_2"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E8:28_1_1_2_2_1"/>
<font color="Maroon" title="c7">m'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</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="E2:28_1_1_2_2"/><b>then </b>
( <font color="Maroon" title="c4">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">k</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E3:28_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E4:28_1_1_2_2"/>
<font color="Maroon" title="c6">f</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 <b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</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="E9:28_1_1_2"/><i><font color="Green" title="E22">A8</font></i>: 
for <font color="Olive" title="b1">m</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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f</font> holds <br/> <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">m</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="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f</font> implies  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">m</font> )</span><br/>

<b>assume </b><a NAME="E1:28_1_1_2_3"/><i><font color="Green" title="E23">A9</font></i>: 
<font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">m</font></span><br/>

<a NAME="E2:28_1_1_2_3"/><i><font color="Green" title="E24">A10</font></i>: 
<font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E1:28_1_1_2_3"><i><font color="Green" title="E23">A9</font></i></a>, <a class="ref" href="euler_1.html#T1" title="EULER_1:th.1">EULER_1:1</a></i>;<br/>
<a NAME="E3:28_1_1_2_3"/><b>then </b>
( <font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k1</font> )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E1:28"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:28_1_1_2_3"/><b>then </b>
( <font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">n</font> &amp;  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c7">m</font>,<font color="Maroon" title="c3">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">m</font> )
 
<b>by </b><i><a class="ref" href="euler_1.html#T1" title="EULER_1:th.1">EULER_1:1</a>, <a class="ref" href="xxreal_0.html#D8" title="XXREAL_0:def.8">XXREAL_0:def 8</a></i>;<br/>
<a NAME="E5:28_1_1_2_3"/><b>then </b>
( <font color="Maroon" title="c6">f</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">m</font> &amp; <font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E7:28_1_1_2"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:28_1_1_2_3"/><b>then </b>
( <font color="Maroon" title="c6">f</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E7:28_1_1_2_3"/><b>then </b><i><font color="Green" title="E25">A11</font></i>: 
( not <font color="Maroon" title="c6">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E8:28_1_1_2_3"/><i><font color="Green" title="E26">A12</font></i>: 
<font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k1</font>
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E2:28_1_1_2_3"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1_1_2_3_1_1"/>
( <font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k1</font> or <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k1</font> )
 </span><b>by </b><i><a class="txt" href="stirl2_1.html#E8:28_1_1_2_3"><i><font color="Green" title="E26">A12</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_2_3_1_1_1"><b>suppose </b></a><a NAME="E1:28_1_1_2_3_1_1_1"/><i><font color="Green" title="E27">A13</font></i>: 
<font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">m</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2_3_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">l'</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">l'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> implies <font color="Maroon" title="c8">l'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/><b>assume </b><a NAME="E1:28_1_1_2_3_1_1_1_1"/><i><font color="Green" title="E28">A14</font></i>: 
<font color="Maroon" title="c8">l'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">l'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><a NAME="E2:28_1_1_2_3_1_1_1_1"/>
<font color="Maroon" title="c8">l'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:28_1_1_2_3_1_1_1_1"><i><font color="Green" title="E28">A14</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/><a NAME="E3:28_1_1_2_3_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c8">l'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c5">n</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 <b>by </b><i><a class="ref" href="stirl2_1.html#T8" target="_self" title="STIRL2_1:th.8">Th8</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c9">l</font> = <font color="Maroon" title="c8">l'</font> as    <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>  ;<br/><a NAME="E5:28_1_1_2_3_1_1_1_1"/>
( <font color="Maroon" title="c9">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c6">f</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c9">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:28_1_1_2_3_1_1_1_1"><i><font color="Green" title="E28">A14</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/><a NAME="E6:28_1_1_2_3_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c9">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c6">f</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c9">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">m</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E7:28_1_1_2_3_1_1_1_1"/><b>then </b>
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c9">l</font>,<font color="Maroon" title="c3">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">m</font>
 <b>by </b><i><a class="txt" href="stirl2_1.html#E7:28_1_1_2"><i><font color="Green" title="E20">A5</font></i></a></i>;<br/><a NAME="E8:28_1_1_2_3_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">m</font>
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:28_1_1_2_3_1_1_1"><i><font color="Green" title="E27">A13</font></i></a>, <a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/><b>hence </b><a NAME="E9:28_1_1_2_3_1_1_1_1"/>
<font color="Maroon" title="c8">l'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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="E3:28_1_1_2_3_1_1_1"/><b>then </b>
( <font color="Maroon" title="c6">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp;  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E7:28_1_1_2_3"><i><font color="Green" title="E25">A11</font></i></a>, <a class="ref" href="nat_1.html#D1" title="NAT_1:def.1">NAT_1:def 1</a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>hence </b><a NAME="E4:28_1_1_2_3_1_1_1"/>
 <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">m</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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="28_1_1_2_3_1_1_2"><b>suppose </b></a><a NAME="E1:28_1_1_2_3_1_1_2"/><i><font color="Green" title="E27">A15</font></i>: 
<font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">m</font></span><br/><div class="add"><a NAME="E2:28_1_1_2_3_1_1_2"/>
for <font color="Olive" title="b1">l</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds <br/><font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">l</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2_3_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">l</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"> ( <font color="Maroon" title="c8">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> implies <font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">l</font> )</span><br/>

<b>assume </b><a NAME="E1:28_1_1_2_3_1_1_2_1"/><i><font color="Green" title="E28">A16</font></i>: 
<font color="Maroon" title="c8">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">l</font></span><br/>

<a NAME="E2:28_1_1_2_3_1_1_2_1"/>
( <font color="Maroon" title="c8">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c6">f</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c8">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E1:28_1_1_2_3_1_1_2_1"><i><font color="Green" title="E28">A16</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:28_1_1_2_3_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c6">f</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c8">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c8">l</font>,<font color="Maroon" title="c7">m</font> &amp; <font color="Maroon" title="c6">f</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c8">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">m</font> )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E7:28_1_1_2"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="stirl2_1.html#E1:28_1_1_2_3_1_1_2"><i><font color="Green" title="E27">A15</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:28_1_1_2_3_1_1_2_1"/>
<font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">l</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D8" title="XXREAL_0:def.8">XXREAL_0:def 8</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>hence </b><a NAME="E3:28_1_1_2_3_1_1_2"/>
 <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">m</font>
 <b>by </b><i><a class="txt" href="stirl2_1.html#E7:28_1_1_2_3"><i><font color="Green" title="E25">A11</font></i></a>, <a class="ref" href="nat_1.html#D1" title="NAT_1:def.1">NAT_1:def 1</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="E10:28_1_1_2_3"/>
 <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">m</font>
 ; <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="hs2(this)" href="javascript:()" title="28_1_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">l</font>, <font color="Maroon" title="c8">m</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="c7">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c8">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c7">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">m</font> implies  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">l</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> )</span><br/><b>assume </b><a NAME="E1:28_1_1_2_4"/><i><font color="Green" title="E23">A17</font></i>: 
( <font color="Maroon" title="c7">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c8">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c7">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">m</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">l</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span><br/><a NAME="E2:28_1_1_2_4"/>
(  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">l</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">l</font> &amp;  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">m</font> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E9:28_1_1_2"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="stirl2_1.html#E1:28_1_1_2_4"><i><font color="Green" title="E23">A17</font></i></a></i>;<br/><b>hence </b><a NAME="E3:28_1_1_2_4"/>
 <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">l</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:28_1_1_2_4"><i><font color="Green" title="E23">A17</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="E11:28_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">f</font> is <a href="stirl2_1.html#V1" title="STIRL2_1:attr.1">"increasing</a>
 <b>by </b><i><a class="ref" href="stirl2_1.html#D1" target="_self" title="STIRL2_1:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E12:28_1_1_2"/>
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font> st <br/>( <font color="Olive" title="b1">f</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> &amp; <font color="Olive" title="b1">f</font> is <a href="stirl2_1.html#V1" title="STIRL2_1:attr.1">"increasing</a> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E8:28_1_1_2"><i><font color="Green" title="E21">A6</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="E3:28"/>
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font> st <br/>( <font color="Olive" title="b1">f</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> &amp; <font color="Olive" title="b1">f</font> is <a href="stirl2_1.html#V1" title="STIRL2_1:attr.1">"increasing</a> )
 ; <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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