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

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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"> for <font color="Olive" title="b1">f</font> being <a href="polynom1.html#V1" title="POLYNOM1:attr.1">finite-support</a> <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">support</a> <font color="Olive" title="b1">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Olive" title="b1">f</font>,1</span>)</span></span><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  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">X</font>] <b>means </b>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">X</font>, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st ( for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st not <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 holds <br/><font color="Olive" title="b1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) holds <br/> <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> $1,<span class="p1">(<span class="default"><a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Olive" title="b1">f</font>,1</span>)</span>;<br/>
<a NAME="E1:125"/><i><font color="Green" title="E72">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c1">X</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>, <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"> ( ( 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="c1">X</font>  st not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c2">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) implies  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c2">f</font>,1</span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:125_1"/><i><font color="Green" title="E73">A2</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="c1">X</font>  st not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c2">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</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">  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c2">f</font>,1</span>)</span></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Maroon" title="c2">f</font> implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> ) &amp; ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Maroon" title="c2">f</font> ) )</span><br/><div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Maroon" title="c2">f</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:125_1_1_1"/><i><font color="Green" title="E74">A3</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Maroon" title="c2">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font></span><br/><b>then reconsider </b><font color="Maroon" title="c4">x'</font> = <font color="Maroon" title="c3">x</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></i>;<br/><a NAME="E3:125_1_1_1"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">x'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c5">x''</font> = <font color="Maroon" title="c4">x'</font> as  <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-valued</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">c</font></span><br/><a NAME="E1:125_1_1_1_1"/><i><font color="Green" title="E75">A4</font></i>: 
<font color="Maroon" title="c5">x''</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">c</font>
 <b>by </b><i><a class="txt" href="#E1:125_1_1_1"><i><font color="Green" title="E74">A3</font></i></a>, <a class="ref" href="polynom1.html#D17" target="_self" title="POLYNOM1:def.17">Def17</a></i>;<br/><a NAME="E2:125_1_1_1_1"/>
<font color="Maroon" title="c2">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E1:125_1"><i><font color="Green" title="E73">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E3:125_1_1_1_1"/>
<font color="Maroon" title="c4">x'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">c</font>
 <b>by </b><i><a class="txt" href="#E1:125_1_1_1_1"><i><font color="Green" title="E75">A4</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>hence </b><a NAME="E6:125_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</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="E2:125_1_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Maroon" title="c2">f</font> )
 <b>by </b><i><a class="ref" href="polynom1.html#T65" target="_self" title="POLYNOM1:th.65">Th65</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="E3:125_1"/><b>then </b>
 <a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">f</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b> <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Maroon" title="c2">f</font></span>)</span> = 
1
<b>by </b><i><a class="ref" href="card_1.html#T50" title="CARD_1:th.50">CARD_1:50</a></i>
<br/>.= 
<a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c2">f</font>,1</span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#T10" title="BINOP_2:th.10">BINOP_2:10</a>, <a class="ref" href="setwiseo.html#T40" title="SETWISEO:th.40">SETWISEO:40</a></i>
;<br/> <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="E2:125"/><i><font color="Green" title="E73">A5</font></i>: 
for <font color="Olive" title="b1">B</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B</font>] &amp; not <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b2">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">B</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">X</font>; <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">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">B</font>] &amp; not <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">B</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b1">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]</span><br/><b>let </b><font color="Maroon" title="c3">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</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="c2">B</font>] &amp; not <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">B</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>] )</span><br/>



<b>assume </b><a NAME="E1:125_2"/><i><font color="Green" title="E74">A6</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">B</font>] &amp; not <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</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="c2">B</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]</span><br/>

<b>let </b><font color="Maroon" title="c4">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>, <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"> ( ( 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="c1">X</font>  st not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span> holds <br/><font color="Maroon" title="c4">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) implies  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c4">f</font>,1</span>)</span> )</span><br/>

<b>assume </b><a NAME="E2:125_2"/><i><font color="Green" title="E75">A7</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="c1">X</font>  st not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/><font color="Maroon" title="c4">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</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">  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c4">f</font>,1</span>)</span></span><br/>

<a NAME="E3:125_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c4">f</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><i><font color="Green" title="E76">A8</font></i>: <span class="p1">(<span class="default"><a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c4">f</font>,1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font> = 
<a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font></span>)</span>,1
<b>by </b><i><a class="ref" href="funcop_1.html#T35" title="FUNCOP_1:th.35">FUNCOP_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="ref" href="binop_2.html#D23" title="BINOP_2:def.23">BINOP_2:def 23</a></i>
;<br/>
<b>set </b><font color="Maroon" title="c5">g</font> = <font color="Maroon" title="c4">f</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<a NAME="E5:125_2"/><i><font color="Green" title="E77">A9</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:125_2"/>
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="c1">X</font>  st not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_2_2"><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="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<b>assume </b><a NAME="E1:125_2_2"/><i><font color="Green" title="E78">A10</font></i>: 
not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="125_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:125_2_2_1"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="125_2_2_1_1"><b>suppose </b></a><a NAME="E1:125_2_2_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><b>hence </b><a NAME="E2:125_2_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E5:125_2"><i><font color="Green" title="E77">A9</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</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="125_2_2_1_2"><b>suppose </b></a><a NAME="E1:125_2_2_1_2"/><i><font color="Green" title="E79">A11</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><div><i><font color="Green" title="E80">A12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="125_2_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:125_2_2_1_2_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:125_2_2_1_2_1"/><b>then </b>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:125_2_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:125_2_2"><i><font color="Green" title="E78">A10</font></i></a>, <a class="txt" href="#E1:125_2_2_1_2"><i><font color="Green" title="E79">A11</font></i></a>, <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><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">x</font> = 
<font color="Maroon" title="c4">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
<b>by </b><i><a class="txt" href="#E1:125_2_2_1_2"><i><font color="Green" title="E79">A11</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="#E2:125_2"><i><font color="Green" title="E75">A7</font></i></a>, <a class="txt" href="#E2:125_2_2_1_2"><i><font color="Green" title="E80">A12</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>
<a NAME="E7:125_2"/><b>then </b><i><font color="Green" title="E78">A13</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c2">B</font>,<span class="p1">(<span class="default"><a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,1</span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:125_2"><i><font color="Green" title="E74">A6</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">ng</font> =  <a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="fraenkel.html#V1" title="FRAENKEL:attr.1">functional</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>set </b><font color="Maroon" title="c7">cng</font> =  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">ng</font>;<br/>
<a NAME="E9:125_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">B</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">B</font>
 
<b>by </b><i><a class="txt" href="#E1:125_2"><i><font color="Green" title="E74">A6</font></i></a>, <a class="ref" href="funct_7.html#T94" title="FUNCT_7:th.94">FUNCT_7:94</a></i>;<br/>
<a NAME="E10:125_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,1</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">B</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c4">f</font>,1</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">B</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T36" title="FUNCOP_1:th.36">FUNCOP_1:36</a></i>;<br/>
<a NAME="E11:125_2"/><b>then </b><i><font color="Green" title="E79">A14</font></i>: 
<a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c2">B</font>,<span class="p1">(<span class="default"><a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c2">B</font>,<span class="p1">(<span class="default"><a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c4">f</font>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="setwop_2.html#T9" title="SETWOP_2:th.9">SETWOP_2:9</a></i>;<br/>
<a NAME="E12:125_2"/><i><font color="Green" title="E80">A15</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">fb1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 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>  ;<br/>
<a NAME="E14:125_2"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c6">ng</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_2_3"><b>proof </b></a><div class="add">

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">ng</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">fb1</font>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = $1 <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">b</font>,$2;<br/>
<a NAME="E1:125_2_3"/><i><font color="Green" title="E81">A16</font></i>: 
for <font color="Olive" title="b1">p</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">ng</font><br/> for <font color="Olive" title="b2">l</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">fb1</font> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">l</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Maroon" title="c4">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">p</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">ng</font>; <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">l</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">fb1</font> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c9">p</font>,<font color="Olive" title="b1">l</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Maroon" title="c4">f</font></span><br/><b>let </b><font color="Maroon" title="c10">l</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">fb1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c10">l</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Maroon" title="c4">f</font></span><br/>



<b>reconsider </b><font color="Maroon" title="c11">q</font> = <font color="Maroon" title="c9">p</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> ;<br/>
<a NAME="E2:125_2_3_1"/><i><font color="Green" title="E82">A17</font></i>: 
<font color="Maroon" title="c8">fb1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
;<br/>
<a NAME="E3:125_2_3_1"/>
<font color="Maroon" title="c10">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">fb1</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">k</font> = <font color="Maroon" title="c10">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>  <b>by </b><i><a class="txt" href="#E2:125_2_3_1"><i><font color="Green" title="E82">A17</font></i></a></i>;<br/>
<a NAME="E5:125_2_3_1"/>
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 
;<br/>
<a NAME="E6:125_2_3_1"/><b>then </b><i><font color="Green" title="E83">A18</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T169" title="FUNCT_2:th.169">FUNCT_2:169</a></i>;<br/>
<a NAME="E7:125_2_3_1"/><b>then </b>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c10">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">pbl</font> = <font color="Maroon" title="c11">q</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c12">k</font> as  <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-valued</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E9:125_2_3_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c13">pbl</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_2_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">x</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="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> implies <font color="Maroon" title="c13">pbl</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c14">x</font> )</span><br/>

<b>assume </b><a NAME="E1:125_2_3_1_1"/><i><font color="Green" title="E84">A19</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">pbl</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="125_2_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:125_2_3_1_1_1"/>
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> or <font color="Maroon" title="c14">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="125_2_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:125_2_3_1_1_1_1"/><i><font color="Green" title="E85">A20</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">pbl</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span><br/><div class="add"><a NAME="E2:125_2_3_1_1_1_1"/><b>then </b><i><font color="Green" title="E86">A21</font></i>: 
<font color="Maroon" title="c13">pbl</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">k</font>
 <b>by </b><i><a class="txt" href="#E6:125_2_3_1"><i><font color="Green" title="E83">A18</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><a NAME="E3:125_2_3_1_1_1_1"/>
<font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">fb1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></i>;<br/><b>hence </b><a NAME="E4:125_2_3_1_1_1_1"/>
<font color="Maroon" title="c13">pbl</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
 <b>by </b><i><a class="txt" href="#E1:125_2_3_1_1_1_1"><i><font color="Green" title="E85">A20</font></i></a>, <a class="txt" href="#E2:125_2_3_1_1_1_1"><i><font color="Green" title="E86">A21</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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="125_2_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:125_2_3_1_1_1_2"/><i><font color="Green" title="E85">A22</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">pbl</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span><br/><div class="add"><a NAME="E2:125_2_3_1_1_1_2"/><b>then </b><i><font color="Green" title="E86">A23</font></i>: 
<font color="Maroon" title="c13">pbl</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><a NAME="E3:125_2_3_1_1_1_2"/>
<font color="Maroon" title="c11">q</font> is    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E6:125_2_3_1"><i><font color="Green" title="E83">A18</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E4:125_2_3_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c11">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
 <b>by </b><i><a class="txt" href="#E1:125_2_3_1_1"><i><font color="Green" title="E84">A19</font></i></a>, <a class="ref" href="polynom1.html#D17" target="_self" title="POLYNOM1:def.17">Def17</a></i>;<br/><b>hence </b><a NAME="E5:125_2_3_1_1_1_2"/>
<font color="Maroon" title="c13">pbl</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
 <b>by </b><i><a class="txt" href="#E1:125_2_3_1_1_1_2"><i><font color="Green" title="E85">A22</font></i></a>, <a class="txt" href="#E2:125_2_3_1_1_1_2"><i><font color="Green" title="E86">A23</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</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="E10:125_2_3_1"/>
<font color="Maroon" title="c9">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c10">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D17" target="_self" title="POLYNOM1:def.17">Def17</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>consider </b><font color="Maroon" title="c9">r</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c6">ng</font>,<font color="Maroon" title="c8">fb1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Maroon" title="c4">f</font><b> such that </b><br/><a NAME="E3:125_2_3"/><i><font color="Green" title="E82">A24</font></i>: 
for <font color="Olive" title="b1">p</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">ng</font><br/> for <font color="Olive" title="b2">l</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">fb1</font> holds  <font color="Maroon" title="c9">r</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">l</font>)
 <b>from </b><i><a class="ref" href="funct_7.html#S1" title="FUNCT_7:sch.1">FUNCT_7:sch 1</a>(<a class="txt" href="#E1:125_2_3"><i><font color="Green" title="E81">A16</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c9">r</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">r</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c6">ng</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <font color="Maroon" title="c9">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Maroon" title="c4">f</font> )</span><br/>

<b>thus </b><a NAME="E4:125_2_3"/>
<font color="Maroon" title="c9">r</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c6">ng</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <font color="Maroon" title="c9">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Maroon" title="c4">f</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_2_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x1</font>, <font color="Maroon" title="c11">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">r</font> or  not <font color="Maroon" title="c11">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">r</font> or  not <font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font> or <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x2</font> )</span><br/>

<b>assume </b><a NAME="E1:125_2_3_2"/><i><font color="Green" title="E83">A25</font></i>: 
( <font color="Maroon" title="c10">x1</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="c9">r</font> &amp; <font color="Maroon" title="c11">x2</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="c9">r</font> &amp; <font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x2</font></span><br/>

<b>then consider </b><font color="Maroon" title="c12">p1</font>, <font color="Maroon" title="c13">l1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:125_2_3_2"/><i><font color="Green" title="E84">A26</font></i>: 
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">p1</font>,<font color="Maroon" title="c13">l1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E4:125_2_3_2"/><i><font color="Green" title="E85">A27</font></i>: 
( <font color="Maroon" title="c12">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">ng</font> &amp; <font color="Maroon" title="c13">l1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">fb1</font> )
 
<b>by </b><i><a class="txt" href="#E1:125_2_3_2"><i><font color="Green" title="E83">A25</font></i></a>, <a class="txt" href="#E3:125_2_3_2"><i><font color="Green" title="E84">A26</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">p2</font>, <font color="Maroon" title="c15">l2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:125_2_3_2"/><i><font color="Green" title="E86">A28</font></i>: 
<font color="Maroon" title="c11">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">p2</font>,<font color="Maroon" title="c15">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="#E1:125_2_3_2"><i><font color="Green" title="E83">A25</font></i></a>, <a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E7:125_2_3_2"/><i><font color="Green" title="E87">A29</font></i>: 
( <font color="Maroon" title="c14">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">ng</font> &amp; <font color="Maroon" title="c15">l2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">fb1</font> )
 
<b>by </b><i><a class="txt" href="#E1:125_2_3_2"><i><font color="Green" title="E83">A25</font></i></a>, <a class="txt" href="#E6:125_2_3_2"><i><font color="Green" title="E86">A28</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">p1</font> = <font color="Maroon" title="c12">p1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <b>by </b><i><a class="txt" href="#E1:125_2_3_2"><i><font color="Green" title="E83">A25</font></i></a>, <a class="txt" href="#E3:125_2_3_2"><i><font color="Green" title="E84">A26</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">p2</font> = <font color="Maroon" title="c14">p2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <b>by </b><i><a class="txt" href="#E1:125_2_3_2"><i><font color="Green" title="E83">A25</font></i></a>, <a class="txt" href="#E6:125_2_3_2"><i><font color="Green" title="E86">A28</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<i><font color="Green" title="E88">A30</font></i>: <font color="Maroon" title="c16">p1</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c13">l1</font> = 
<font color="Maroon" title="c9">r</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c16">p1</font>,<font color="Maroon" title="c13">l1</font>
<b>by </b><i><a class="txt" href="#E3:125_2_3"><i><font color="Green" title="E82">A24</font></i></a>, <a class="txt" href="#E4:125_2_3_2"><i><font color="Green" title="E85">A27</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c9">r</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c17">p2</font>,<font color="Maroon" title="c15">l2</font>
<b>by </b><i><a class="txt" href="#E1:125_2_3_2"><i><font color="Green" title="E83">A25</font></i></a>, <a class="txt" href="#E3:125_2_3_2"><i><font color="Green" title="E84">A26</font></i></a>, <a class="txt" href="#E6:125_2_3_2"><i><font color="Green" title="E86">A28</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c17">p2</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c15">l2</font>
<b>by </b><i><a class="txt" href="#E3:125_2_3"><i><font color="Green" title="E82">A24</font></i></a>, <a class="txt" href="#E7:125_2_3_2"><i><font color="Green" title="E87">A29</font></i></a></i>
;<br/>
<a NAME="E11:125_2_3_2"/><i><font color="Green" title="E89">A31</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c16">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T169" title="FUNCT_2:th.169">FUNCT_2:169</a></i>;<br/>
<a NAME="E12:125_2_3_2"/><i><font color="Green" title="E90">A32</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c17">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T169" title="FUNCT_2:th.169">FUNCT_2:169</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c18">p1'</font> = <font color="Maroon" title="c16">p1</font>, <font color="Maroon" title="c19">p2'</font> = <font color="Maroon" title="c17">p2</font> as  <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-valued</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="#E11:125_2_3_2"><i><font color="Green" title="E89">A31</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<div><i><font color="Green" title="E91">A33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="125_2_3_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">x</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="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> implies <font color="Maroon" title="c18">p1'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">p2'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><b>assume </b><a NAME="E1:125_2_3_2_2"/><i><font color="Green" title="E92">A34</font></i>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c18">p1'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">p2'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="125_2_3_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:125_2_3_2_2_1"/>
( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> or <font color="Maroon" title="c20">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="125_2_3_2_2_1_1"><b>suppose </b></a><a NAME="E1:125_2_3_2_2_1_1"/><i><font color="Green" title="E93">A35</font></i>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c18">p1'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">p2'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:125_2_3_2_2_1_1"/><i><font color="Green" title="E94">A36</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E5:125_2"><i><font color="Green" title="E77">A9</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><a NAME="E3:125_2_3_2_2_1_1"/><i><font color="Green" title="E95">A37</font></i>: 
( <font color="Maroon" title="c18">p1'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">x</font> &amp; <font color="Maroon" title="c19">p2'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">x</font> )
 <b>by </b><i><a class="txt" href="#E1:125_2_3_2_2"><i><font color="Green" title="E92">A34</font></i></a>, <a class="ref" href="polynom1.html#D17" target="_self" title="POLYNOM1:def.17">Def17</a></i>;<br/><b>hence </b><font color="Maroon" title="c18">p1'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">x</font> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="#E1:125_2_3_2_2_1_1"><i><font color="Green" title="E93">A35</font></i></a>, <a class="txt" href="#E2:125_2_3_2_2_1_1"><i><font color="Green" title="E94">A36</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c19">p2'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">x</font>
<b>by </b><i><a class="txt" href="#E1:125_2_3_2_2_1_1"><i><font color="Green" title="E93">A35</font></i></a>, <a class="txt" href="#E2:125_2_3_2_2_1_1"><i><font color="Green" title="E94">A36</font></i></a>, <a class="txt" href="#E3:125_2_3_2_2_1_1"><i><font color="Green" title="E95">A37</font></i></a></i>
;<br/> <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="125_2_3_2_2_1_2"><b>suppose </b></a><a NAME="E1:125_2_3_2_2_1_2"/><i><font color="Green" title="E93">A38</font></i>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c18">p1'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">p2'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><b>hence </b><font color="Maroon" title="c18">p1'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">p1</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c13">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font>
<b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>
<br/>.= 
<font color="Maroon" title="c19">p2'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">x</font>
<b>by </b><i><a class="txt" href="#E10:125_2_3_2"><i><font color="Green" title="E88">A30</font></i></a>, <a class="txt" href="#E1:125_2_3_2_2_1_2"><i><font color="Green" title="E93">A38</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>
;<br/> <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>
<font color="Maroon" title="c13">l1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">p1</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c13">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">b</font>
<b>by </b><i><a class="txt" href="#E11:125_2_3_2"><i><font color="Green" title="E89">A31</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>
<br/>.= 
<font color="Maroon" title="c15">l2</font>
<b>by </b><i><a class="txt" href="#E10:125_2_3_2"><i><font color="Green" title="E88">A30</font></i></a>, <a class="txt" href="#E12:125_2_3_2"><i><font color="Green" title="E90">A32</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>
;<br/>
<b>hence </b><a NAME="E16:125_2_3_2"/>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x2</font>
 <b>by </b><i><a class="txt" href="#E3:125_2_3_2"><i><font color="Green" title="E84">A26</font></i></a>, <a class="txt" href="#E6:125_2_3_2"><i><font color="Green" title="E86">A28</font></i></a>, <a class="txt" href="#E14:125_2_3_2"><i><font color="Green" title="E91">A33</font></i></a>, <a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE: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>

<b>thus </b><a NAME="E5:125_2_3"/><i><font color="Green" title="E83">A39</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c6">ng</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <font color="Maroon" title="c9">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Maroon" title="c4">f</font></span><br/>

<b>thus </b><a NAME="E6:125_2_3"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Maroon" title="c4">f</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <font color="Maroon" title="c9">r</font></span><br/>

<b>thus </b><a NAME="E7:125_2_3"/>
 <a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Maroon" title="c4">f</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="c9">r</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="125_2_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Maroon" title="c4">f</font> or <font color="Maroon" title="c10">x</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="c9">r</font> )</span><br/>

<b>assume </b><a NAME="E1:125_2_3_3"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Maroon" title="c4">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">x</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="c9">r</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c11">e</font> = <font color="Maroon" title="c10">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Maroon" title="c4">f</font> ;<br/>
<a NAME="E3:125_2_3_3"/><i><font color="Green" title="E84">A40</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T169" title="FUNCT_2:th.169">FUNCT_2:169</a></i>;<br/>
<a NAME="E4:125_2_3_3"/><b>then </b><i><font color="Green" title="E85">A41</font></i>: 
<font color="Maroon" title="c11">e</font> is    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:125_2_3_3"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">e</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="#E3:125_2_3_3"><i><font color="Green" title="E84">A40</font></i></a>, <a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">eb0</font> = <font color="Maroon" title="c11">e</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  as  <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-valued</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_2_3_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">x</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="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> implies <font color="Maroon" title="c12">eb0</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><b>assume </b><a NAME="E1:125_2_3_3_1"/><i><font color="Green" title="E86">A42</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">eb0</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="125_2_3_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:125_2_3_3_1_1"/>
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> or <font color="Maroon" title="c13">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="125_2_3_3_1_1_1"><b>suppose </b></a><a NAME="E1:125_2_3_3_1_1_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">eb0</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><b>hence </b><a NAME="E2:125_2_3_3_1_1_1"/>
<font color="Maroon" title="c12">eb0</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">x</font>
 <b>by </b><i><a class="txt" href="#E3:125_2_3_3"><i><font color="Green" title="E84">A40</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</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="125_2_3_3_1_1_2"><b>suppose </b></a><a NAME="E1:125_2_3_3_1_1_2"/><i><font color="Green" title="E87">A43</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">eb0</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:125_2_3_3_1_1_2"/><b>then </b><i><font color="Green" title="E88">A44</font></i>: 
<font color="Maroon" title="c12">eb0</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">x</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><a NAME="E3:125_2_3_3_1_1_2"/>
<font color="Maroon" title="c11">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">x</font>
 <b>by </b><i><a class="txt" href="#E4:125_2_3_3"><i><font color="Green" title="E85">A41</font></i></a>, <a class="txt" href="#E1:125_2_3_3_1"><i><font color="Green" title="E86">A42</font></i></a>, <a class="ref" href="polynom1.html#D17" target="_self" title="POLYNOM1:def.17">Def17</a></i>;<br/><b>hence </b><a NAME="E4:125_2_3_3_1_1_2"/>
<font color="Maroon" title="c12">eb0</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">x</font>
 <b>by </b><i><a class="txt" href="#E1:125_2_3_3_1_1_2"><i><font color="Green" title="E87">A43</font></i></a>, <a class="txt" href="#E2:125_2_3_3_1_1_2"><i><font color="Green" title="E88">A44</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</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>then reconsider </b><font color="Maroon" title="c13">eb0</font> = <font color="Maroon" title="c12">eb0</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D17" target="_self" title="POLYNOM1:def.17">Def17</a></i>;<br/>
<a NAME="E9:125_2_3_3"/>
<font color="Maroon" title="c11">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="txt" href="#E4:125_2_3_3"><i><font color="Green" title="E85">A41</font></i></a>, <a class="ref" href="polynom1.html#D17" target="_self" title="POLYNOM1:def.17">Def17</a></i>;<br/>
<a NAME="E10:125_2_3_3"/><b>then </b>
<font color="Maroon" title="c11">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">fb1</font>
 
<b>by </b><i><a class="txt" href="#E12:125_2"><i><font color="Green" title="E80">A15</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:125_2_3_3"/><b>then </b><i><font color="Green" title="E86">A45</font></i>: 
<font color="Maroon" title="c11">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">fb1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></i>;<br/>
<a NAME="E12:125_2_3_3"/><b>then </b><i><font color="Green" title="E87">A46</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">eb0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c9">r</font>
 
<b>by </b><i><a class="txt" href="#E5:125_2_3"><i><font color="Green" title="E83">A39</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<font color="Maroon" title="c11">e</font> = 
<font color="Maroon" title="c11">e</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">b</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#T37" title="FUNCT_7:th.37">FUNCT_7:37</a></i>
<br/>.= 
<font color="Maroon" title="c13">eb0</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">b</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#T36" title="FUNCT_7:th.36">FUNCT_7:36</a></i>
;<br/>
<a NAME="E14:125_2_3_3"/><b>then </b>
<font color="Maroon" title="c11">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">eb0</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:125_2_3"><i><font color="Green" title="E82">A24</font></i></a>, <a class="txt" href="#E11:125_2_3_3"><i><font color="Green" title="E86">A45</font></i></a></i>;<br/>
<b>hence </b><a NAME="E15:125_2_3_3"/>
<font color="Maroon" title="c10">x</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="c9">r</font>
 <b>by </b><i><a class="txt" href="#E12:125_2_3_3"><i><font color="Green" title="E87">A46</font></i></a>, <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>


</div><b>end;</b></div>
<b>hence </b> <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Maroon" title="c4">f</font></span>)</span> = 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c6">ng</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">ng</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="card_2.html#T65" title="CARD_2:th.65">CARD_2:65</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">ng</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></i>
<br/>.= 
<a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c2">B</font>,<span class="p2">(<span class="default"><a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c4">f</font>,1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="#E7:125_2"><i><font color="Green" title="E78">A13</font></i></a>, <a class="txt" href="#E11:125_2"><i><font color="Green" title="E79">A14</font></i></a>, <a class="ref" href="binop_2.html#D24" title="BINOP_2:def.24">BINOP_2:def 24</a></i>
<br/>.= 
<a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c4">f</font>,1</span>)</span>
<b>by </b><i><a class="txt" href="#E1:125_2"><i><font color="Green" title="E74">A6</font></i></a>, <a class="txt" href="#E4:125_2"><i><font color="Green" title="E76">A8</font></i></a>, <a class="ref" href="setwop_2.html#T4" title="SETWOP_2:th.4">SETWOP_2:4</a></i>
;<br/> <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="E3:125"/><i><font color="Green" title="E74">A47</font></i>: 
for <font color="Olive" title="b1">B</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">X</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B</font>]
 
<b>from </b><i><a class="ref" href="setwiseo.html#S2" title="SETWISEO:sch.2">SETWISEO:sch 2</a>(<a class="txt" href="#E1:125"><i><font color="Green" title="E72">A1</font></i></a>, <a class="txt" href="#E2:125"><i><font color="Green" title="E73">A5</font></i></a>);<br/></i>
<b>let </b><font color="Maroon" title="c2">f</font> be  <a href="polynom1.html#V1" title="POLYNOM1:attr.1">finite-support</a> <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>, <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">  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">support</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c2">f</font>,1</span>)</span></span><br/>

<a NAME="E4:125"/>
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="c1">X</font>  st not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K17" title="POLYNOM1:func.17">support</a> <font color="Maroon" title="c2">f</font> holds <br/><font color="Maroon" title="c2">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="polynom1.html#D7" target="_self" title="POLYNOM1:def.7">Def7</a></i>;<br/>
<b>hence </b><a NAME="E5:125"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">support</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c2">f</font>,1</span>)</span>
 <b>by </b><i><a class="txt" href="#E3:125"><i><font color="Green" title="E74">A47</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>
