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

<b>let </b><font color="Maroon" title="c1">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Fy</font> being <a href="finset_1.html#V2" title="FINSET_1:attr.2">finite-yielding</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><br/> for <font color="Olive" title="b2">X</font> being <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/> for <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b4">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">Fy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">X</font> &amp; <font color="Olive" title="b3">x</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="Olive" title="b1">Fy</font> &amp; <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> <a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <font color="Olive" title="b1">Fy</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">Fy</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <span class="p2">(<span class="default"><a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">Fy</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">Fy</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">k</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">Fy</font> be  <a href="finset_1.html#V2" title="FINSET_1:attr.2">finite-yielding</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</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="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b3">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b2">x</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="c2">Fy</font> &amp; <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> <a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <font color="Maroon" title="c2">Fy</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <span class="p2">(<span class="default"><a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">k</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">X</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</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">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b1">x</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="c2">Fy</font> &amp; <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> <a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <font color="Maroon" title="c2">Fy</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <span class="p2">(<span class="default"><a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">k</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">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"> for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c4">x</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="c2">Fy</font> &amp; <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> <a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <font color="Maroon" title="c2">Fy</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <span class="p2">(<span class="default"><a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">k</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c4">x</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="c2">Fy</font> &amp; <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <font color="Maroon" title="c2">Fy</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <span class="p2">(<span class="default"><a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">k</font></span>)</span> )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:73"/><i><font color="Green" title="E57">A1</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c4">x</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="c2">Fy</font> )
 <b>and </b><br/><a NAME="E2:73"/><i><font color="Green" title="E58">A2</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <font color="Maroon" title="c2">Fy</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <span class="p2">(<span class="default"><a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">k</font></span>)</span></span><br/>

<b>set </b><font color="Maroon" title="c6">Xx</font> = <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<b>set </b><font color="Maroon" title="c7">X0</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c8">X1</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span>}</span> ;<br/>
<a NAME="E3:73"/>
not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/>
<a NAME="E4:73"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="card_fin.html#E18"><i><font color="Green" title="E16">Lm1</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">X0</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> , <font color="Maroon" title="c10">X1</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span>}</span>  as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">P0</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">X0</font>,<font color="Maroon" title="c9">X0</font><b> such that </b><br/><a NAME="E7:73"/><i><font color="Green" title="E59">A3</font></i>: 
<font color="Maroon" title="c11">P0</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="card_fin.html#E54"><i><font color="Green" title="E45">Lm3</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">P0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 &amp; <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X0</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> );<br/>
<a NAME="E8:73"/><i><font color="Green" title="E60">A4</font></i>: 
for <font color="Olive" title="b1">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">X0</font> holds <br/> ex <font color="Olive" title="b2">P0x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">P0x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x0</font>,<font color="Olive" title="b2">P0x0</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x0</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="c12">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">X0</font> implies  ex <font color="Olive" title="b1">P0x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">P0x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">x0</font>,<font color="Olive" title="b1">P0x0</font>] ) )</span><br/>

<b>assume </b><a NAME="E1:73_1"/><i><font color="Green" title="E61">A5</font></i>: 
<font color="Maroon" title="c12">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">X0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">P0x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">P0x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">x0</font>,<font color="Olive" title="b1">P0x0</font>] )</span><br/>

<a NAME="E2:73_1"/>
<font color="Maroon" title="c12">x0</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="c11">P0</font>
 
<b>by </b><i><a class="txt" href="#E1:73_1"><i><font color="Green" title="E61">A5</font></i></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:73_1"/><b>then </b><i><font color="Green" title="E62">A6</font></i>: 
<font color="Maroon" title="c11">P0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x0</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="c11">P0</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:73_1"/><b>then </b>
<font color="Maroon" title="c11">P0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X0</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c13">P0x0</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span><b> such that </b><br/><a NAME="E6:73_1"/><i><font color="Green" title="E63">A7</font></i>: 
( <font color="Maroon" title="c11">P0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">P0x0</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">P0x0</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; <font color="Maroon" title="c13">P0x0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/>
<a NAME="E7:73_1"/>
<font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E8:73_1"/><b>then </b><i><font color="Green" title="E64">A8</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">P0x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">P0x0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <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="c13">P0x0</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relat_1.html#T99" title="RELAT_1:th.99">RELAT_1:99</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E9:73_1"/><b>then </b>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">P0x0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">P0x0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">Px</font> = <font color="Maroon" title="c13">P0x0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <b>by </b><i><a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/>
<a NAME="E11:73_1"/>
not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/>
<a NAME="E12:73_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T141" title="ZFMISC_1:th.141">ZFMISC_1:141</a></i>;<br/>
<a NAME="E13:73_1"/><b>then </b>
<font color="Maroon" title="c13">P0x0</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">Px</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="#E6:73_1"><i><font color="Green" title="E63">A7</font></i></a>, <a class="txt" href="#E8:73_1"><i><font color="Green" title="E64">A8</font></i></a>, <a class="ref" href="card_fin.html#T15" target="_self" title="CARD_FIN:th.15">Th15</a></i>;<br/>
<a NAME="E14:73_1"/><b>then </b>
<font color="Maroon" title="c14">Px</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E6:73_1"><i><font color="Green" title="E63">A7</font></i></a>, <a class="ref" href="card_fin.html#D1" target="_self" title="CARD_FIN:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E15:73_1"/>
 ex <font color="Olive" title="b1">P0x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">P0x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">x0</font>,<font color="Olive" title="b1">P0x0</font>] )
 <b>by </b><i><a class="txt" href="#E3:73_1"><i><font color="Green" title="E62">A6</font></i></a>, <a class="txt" href="#E6:73_1"><i><font color="Green" title="E63">A7</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>consider </b><font color="Maroon" title="c12">P0x</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">X0</font>, <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <b> such that </b><br/><a NAME="E10:73"/><i><font color="Green" title="E61">A9</font></i>: 
for <font color="Olive" title="b1">x1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">X0</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x1</font>,<font color="Maroon" title="c12">P0x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="#E8:73"><i><font color="Green" title="E60">A4</font></i></a>);<br/></i>
<a NAME="E11:73"/>
not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/>
<a NAME="E12:73"/><b>then </b><i><font color="Green" title="E62">A10</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">X0</font>
 
<b>by </b><i><a class="ref" href="card_fin.html#T16" target="_self" title="CARD_FIN:th.16">Th16</a></i>;<br/>
<a NAME="E13:73"/>
for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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="c12">P0x</font> &amp; <font color="Olive" title="b2">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="c12">P0x</font> &amp; <font color="Maroon" title="c12">P0x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">P0x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x1</font>, <font color="Maroon" title="c14">x2</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">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="c12">P0x</font> &amp; <font color="Maroon" title="c14">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="c12">P0x</font> &amp; <font color="Maroon" title="c12">P0x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">P0x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x2</font> implies <font color="Maroon" title="c13">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x2</font> )</span><br/>



<b>assume </b><a NAME="E1:73_2"/><i><font color="Green" title="E63">A11</font></i>: 
( <font color="Maroon" title="c13">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="c12">P0x</font> &amp; <font color="Maroon" title="c14">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="c12">P0x</font> &amp; <font color="Maroon" title="c12">P0x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">P0x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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="c13">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x2</font></span><br/>

<a NAME="E2:73_2"/>
( (  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; ( <font color="Maroon" title="c9">X0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">X0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )
 
;<br/>
<a NAME="E3:73_2"/><b>then </b><i><font color="Green" title="E64">A12</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">P0x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">X0</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">P0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">X0</font> )
 
<b>by </b><i><a class="txt" href="#E12:73"><i><font color="Green" title="E62">A10</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">f1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:73_2"/><i><font color="Green" title="E65">A13</font></i>: 
( <font color="Maroon" title="c15">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">P0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x1</font> &amp; <font color="Maroon" title="c15">f1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X0</font> &amp; <font color="Maroon" title="c12">P0x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">f1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E10:73"><i><font color="Green" title="E61">A9</font></i></a>, <a class="txt" href="#E1:73_2"><i><font color="Green" title="E63">A11</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">f2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E7:73_2"/><i><font color="Green" title="E66">A14</font></i>: 
( <font color="Maroon" title="c16">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">P0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x2</font> &amp; <font color="Maroon" title="c16">f2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X0</font> &amp; <font color="Maroon" title="c12">P0x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E10:73"><i><font color="Green" title="E61">A9</font></i></a>, <a class="txt" href="#E1:73_2"><i><font color="Green" title="E63">A11</font></i></a></i>;<br/>
<a NAME="E8:73_2"/><i><font color="Green" title="E67">A15</font></i>: 
(  ex <font color="Olive" title="b1">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> st <br/>( <font color="Maroon" title="c15">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; <font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp;  ex <font color="Olive" title="b1">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> st <br/>( <font color="Maroon" title="c16">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; <font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">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="#E5:73_2"><i><font color="Green" title="E65">A13</font></i></a>, <a class="txt" href="#E7:73_2"><i><font color="Green" title="E66">A14</font></i></a></i>;<br/>
<a NAME="E9:73_2"/><b>then </b><i><font color="Green" title="E68">A16</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c16">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">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="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E10:73_2"/>
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</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="c15">f1</font> holds <br/><font color="Maroon" title="c15">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">z</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="c17">z</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="c15">f1</font> implies <font color="Maroon" title="c15">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> )</span><br/>

<b>assume </b><a NAME="E1:73_2_1"/><i><font color="Green" title="E69">A17</font></i>: 
<font color="Maroon" title="c17">z</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="c15">f1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c15">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:73_2_1_1_1"/>
( <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c17">z</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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="#E9:73_2"><i><font color="Green" title="E68">A16</font></i></a>, <a class="txt" href="#E1:73_2_1"><i><font color="Green" title="E69">A17</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:73_2_1_1_1_1"/>
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</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"> <font color="Maroon" title="c15">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font></span><br/><div class="add"><a NAME="E2:73_2_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> &amp; <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c16">f2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E9:73_2"><i><font color="Green" title="E68">A16</font></i></a>, <a class="txt" href="#E1:73_2_1"><i><font color="Green" title="E69">A17</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:73_2_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c15">f1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c16">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T71" title="FUNCT_1:th.71">FUNCT_1:71</a></i>;<br/><b>hence </b><a NAME="E4:73_2_1_1_1_1"/>
<font color="Maroon" title="c15">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font>
 <b>by </b><i><a class="txt" href="#E1:73_2"><i><font color="Green" title="E63">A11</font></i></a>, <a class="txt" href="#E5:73_2"><i><font color="Green" title="E65">A13</font></i></a>, <a class="txt" href="#E7:73_2"><i><font color="Green" title="E66">A14</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:73_2_1_1_1_2"/>
<font color="Maroon" title="c17">z</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="c4">x</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"> <font color="Maroon" title="c15">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font></span><br/><div class="add"><a NAME="E2:73_2_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 <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 NAME="E3:73_2_1_1_1_2"/>
<font color="Maroon" title="c15">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font>
 <b>by </b><i><a class="txt" href="#E8:73_2"><i><font color="Green" title="E67">A15</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:73_2_1"/>
<font color="Maroon" title="c15">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E11:73_2"/><b>then </b>
( <font color="Maroon" title="c15">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">f2</font> &amp; <font color="Maroon" title="c13">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="c11">P0</font> &amp; <font color="Maroon" title="c14">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="c11">P0</font> )
 
<b>by </b><i><a class="txt" href="#E1:73_2"><i><font color="Green" title="E63">A11</font></i></a>, <a class="txt" href="#E3:73_2"><i><font color="Green" title="E64">A12</font></i></a>, <a class="txt" href="#E9:73_2"><i><font color="Green" title="E68">A16</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E12:73_2"/>
<font color="Maroon" title="c13">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x2</font>
 <b>by </b><i><a class="txt" href="#E7:73"><i><font color="Green" title="E59">A3</font></i></a>, <a class="txt" href="#E5:73_2"><i><font color="Green" title="E65">A13</font></i></a>, <a class="txt" href="#E7:73_2"><i><font color="Green" title="E66">A14</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E14:73"/><b>then </b><i><font color="Green" title="E63">A18</font></i>: 
<font color="Maroon" title="c12">P0x</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E15:73"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E16:73"/><b>then </b><i><font color="Green" title="E64">A19</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="#E1:73"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">XFS0</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E18:73"/><i><font color="Green" title="E65">A20</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c13">XFS0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">P0x</font>
 <b>and </b><br/><a NAME="E19:73"/><i><font color="Green" title="E66">A21</font></i>: 
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c13">XFS0</font> &amp; <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">P0x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> holds <br/><font color="Maroon" title="c13">XFS0</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<font color="Olive" title="b2">f</font>,1</span>)</span>
 <b>and </b><br/><a NAME="E20:73"/><i><font color="Green" title="E67">A22</font></i>: 
 <a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c13">XFS0</font>
 <b>by </b><i><a class="txt" href="#E12:73"><i><font color="Green" title="E62">A10</font></i></a>, <a class="txt" href="#E14:73"><i><font color="Green" title="E63">A18</font></i></a>, <a class="ref" href="card_fin.html#D4" target="_self" title="CARD_FIN:def.4">Def4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">P1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">X1</font>,<font color="Maroon" title="c10">X1</font><b> such that </b><br/><a NAME="E22:73"/><i><font color="Green" title="E68">A23</font></i>: 
<font color="Maroon" title="c14">P1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="card_fin.html#E54"><i><font color="Green" title="E45">Lm3</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 &amp; <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">X1</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> );<br/>
<a NAME="E23:73"/><i><font color="Green" title="E69">A24</font></i>: 
for <font color="Olive" title="b1">x1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">X1</font> holds <br/> ex <font color="Olive" title="b2">P1x1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">P1x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c1">k</font>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">P1x1</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">x1</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="c15">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">X1</font> implies  ex <font color="Olive" title="b1">P1x1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">P1x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c1">k</font>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">x1</font>,<font color="Olive" title="b1">P1x1</font>] ) )</span><br/>

<b>assume </b><a NAME="E1:73_3"/><i><font color="Green" title="E70">A25</font></i>: 
<font color="Maroon" title="c15">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">X1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">P1x1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">P1x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c1">k</font>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">x1</font>,<font color="Olive" title="b1">P1x1</font>] )</span><br/>

<a NAME="E2:73_3"/>
<font color="Maroon" title="c15">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="c14">P1</font>
 
<b>by </b><i><a class="txt" href="#E1:73_3"><i><font color="Green" title="E70">A25</font></i></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:73_3"/><b>then </b><i><font color="Green" title="E71">A26</font></i>: 
<font color="Maroon" title="c14">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x1</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="c14">P1</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:73_3"/><b>then </b>
<font color="Maroon" title="c14">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">X1</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c16">P1x1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span><b> such that </b><br/><a NAME="E6:73_3"/><i><font color="Green" title="E72">A27</font></i>: 
( <font color="Maroon" title="c14">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">P1x1</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">P1x1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; <font color="Maroon" title="c16">P1x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ;<br/>
<a NAME="E7:73_3"/>
<font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E8:73_3"/><b>then </b><i><font color="Green" title="E73">A28</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c16">P1x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">P1x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <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="c16">P1x1</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relat_1.html#T99" title="RELAT_1:th.99">RELAT_1:99</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E9:73_3"/><b>then </b>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">P1x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">P1x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c17">Px</font> = <font color="Maroon" title="c16">P1x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <b>by </b><i><a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/>
<a NAME="E11:73_3"/>
( not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c4">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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c16">P1x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/>
<a NAME="E12:73_3"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c4">x</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="c16">P1x1</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a>, <a class="ref" href="zfmisc_1.html#T141" title="ZFMISC_1:th.141">ZFMISC_1:141</a></i>;<br/>
<a NAME="E13:73_3"/><b>then </b><i><font color="Green" title="E74">A29</font></i>: 
<font color="Maroon" title="c16">P1x1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">Px</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="#E6:73_3"><i><font color="Green" title="E72">A27</font></i></a>, <a class="txt" href="#E8:73_3"><i><font color="Green" title="E73">A28</font></i></a>, <a class="ref" href="card_fin.html#T13" target="_self" title="CARD_FIN:th.13">Th13</a></i>;<br/>
<a NAME="E14:73_3"/>
not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">Px</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/>
<a NAME="E15:73_3"/><b>then </b>
( <font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><font color="Maroon" title="c17">Px</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> )
 
<b>by </b><i><a class="txt" href="#E6:73_3"><i><font color="Green" title="E72">A27</font></i></a>, <a class="txt" href="#E13:73_3"><i><font color="Green" title="E74">A29</font></i></a>, <a class="ref" href="card_2.html#T54" title="CARD_2:th.54">CARD_2:54</a></i>;<br/>
<a NAME="E16:73_3"/><b>then </b>
<font color="Maroon" title="c17">Px</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c1">k</font>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="card_fin.html#D1" target="_self" title="CARD_FIN:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E17:73_3"/>
 ex <font color="Olive" title="b1">P1x1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">P1x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c1">k</font>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">x1</font>,<font color="Olive" title="b1">P1x1</font>] )
 <b>by </b><i><a class="txt" href="#E3:73_3"><i><font color="Green" title="E71">A26</font></i></a>, <a class="txt" href="#E6:73_3"><i><font color="Green" title="E72">A27</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>consider </b><font color="Maroon" title="c15">P1x</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">X1</font>, <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c1">k</font>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <b> such that </b><br/><a NAME="E25:73"/><i><font color="Green" title="E70">A30</font></i>: 
for <font color="Olive" title="b1">x1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">X1</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x1</font>,<font color="Maroon" title="c15">P1x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="#E23:73"><i><font color="Green" title="E69">A24</font></i></a>);<br/></i>
<a NAME="E26:73"/>
not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/>
<a NAME="E27:73"/><b>then </b><i><font color="Green" title="E71">A31</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c1">k</font>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">X1</font>
 
<b>by </b><i><a class="ref" href="card_fin.html#T14" target="_self" title="CARD_FIN:th.14">Th14</a></i>;<br/>
<a NAME="E28:73"/>
for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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="c15">P1x</font> &amp; <font color="Olive" title="b2">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="c15">P1x</font> &amp; <font color="Maroon" title="c15">P1x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">P1x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">x1</font>, <font color="Maroon" title="c17">x2</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="c16">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="c15">P1x</font> &amp; <font color="Maroon" title="c17">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="c15">P1x</font> &amp; <font color="Maroon" title="c15">P1x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">P1x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x2</font> implies <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x2</font> )</span><br/>



<b>assume </b><a NAME="E1:73_4"/><i><font color="Green" title="E72">A32</font></i>: 
( <font color="Maroon" title="c16">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="c15">P1x</font> &amp; <font color="Maroon" title="c17">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="c15">P1x</font> &amp; <font color="Maroon" title="c15">P1x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">P1x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">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="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x2</font></span><br/>

<a NAME="E2:73_4"/>
( (  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c1">k</font>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c1">k</font>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; ( <font color="Maroon" title="c10">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )
 
;<br/>
<a NAME="E3:73_4"/><b>then </b><i><font color="Green" title="E73">A33</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">P1x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">X1</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">X1</font> )
 
<b>by </b><i><a class="txt" href="#E27:73"><i><font color="Green" title="E71">A31</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">f1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:73_4"/><i><font color="Green" title="E74">A34</font></i>: 
( <font color="Maroon" title="c18">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font> &amp; <font color="Maroon" title="c18">f1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">X1</font> &amp; <font color="Maroon" title="c15">P1x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E25:73"><i><font color="Green" title="E70">A30</font></i></a>, <a class="txt" href="#E1:73_4"><i><font color="Green" title="E72">A32</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c19">f2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E7:73_4"/><i><font color="Green" title="E75">A35</font></i>: 
( <font color="Maroon" title="c19">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x2</font> &amp; <font color="Maroon" title="c19">f2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">X1</font> &amp; <font color="Maroon" title="c15">P1x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E25:73"><i><font color="Green" title="E70">A30</font></i></a>, <a class="txt" href="#E1:73_4"><i><font color="Green" title="E72">A32</font></i></a></i>;<br/>
<a NAME="E8:73_4"/><i><font color="Green" title="E76">A36</font></i>: 
(  ex <font color="Olive" title="b1">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> st <br/>( <font color="Maroon" title="c18">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; <font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp;  ex <font color="Olive" title="b1">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> st <br/>( <font color="Maroon" title="c19">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; <font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 
<b>by </b><i><a class="txt" href="#E5:73_4"><i><font color="Green" title="E74">A34</font></i></a>, <a class="txt" href="#E7:73_4"><i><font color="Green" title="E75">A35</font></i></a></i>;<br/>
<a NAME="E9:73_4"/><b>then </b><i><font color="Green" title="E77">A37</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c18">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c19">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c19">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<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="E10:73_4"/>
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</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="c18">f1</font> holds <br/><font color="Maroon" title="c18">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">z</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">z</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="c18">f1</font> implies <font color="Maroon" title="c18">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font> )</span><br/>

<b>assume </b><a NAME="E1:73_4_1"/><i><font color="Green" title="E78">A38</font></i>: 
<font color="Maroon" title="c20">z</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="c18">f1</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">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_4_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_4_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:73_4_1_1_1"/>
( <font color="Maroon" title="c20">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c20">z</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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="#E9:73_4"><i><font color="Green" title="E77">A37</font></i></a>, <a class="txt" href="#E1:73_4_1"><i><font color="Green" title="E78">A38</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_4_1_1_1_1"><b>suppose </b></a><a NAME="E1:73_4_1_1_1_1"/>
<font color="Maroon" title="c20">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</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"> <font color="Maroon" title="c18">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font></span><br/><div class="add"><a NAME="E2:73_4_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c20">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> &amp; <font color="Maroon" title="c20">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c19">f2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E9:73_4"><i><font color="Green" title="E77">A37</font></i></a>, <a class="txt" href="#E1:73_4_1"><i><font color="Green" title="E78">A38</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:73_4_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c18">f1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c19">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T71" title="FUNCT_1:th.71">FUNCT_1:71</a></i>;<br/><b>hence </b><a NAME="E4:73_4_1_1_1_1"/>
<font color="Maroon" title="c18">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font>
 <b>by </b><i><a class="txt" href="#E1:73_4"><i><font color="Green" title="E72">A32</font></i></a>, <a class="txt" href="#E5:73_4"><i><font color="Green" title="E74">A34</font></i></a>, <a class="txt" href="#E7:73_4"><i><font color="Green" title="E75">A35</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_4_1_1_1_2"><b>suppose </b></a><a NAME="E1:73_4_1_1_1_2"/>
<font color="Maroon" title="c20">z</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="c4">x</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"> <font color="Maroon" title="c18">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font></span><br/><div class="add"><a NAME="E2:73_4_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c20">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 <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 NAME="E3:73_4_1_1_1_2"/>
<font color="Maroon" title="c18">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font>
 <b>by </b><i><a class="txt" href="#E8:73_4"><i><font color="Green" title="E76">A36</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:73_4_1"/>
<font color="Maroon" title="c18">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E11:73_4"/><b>then </b>
( <font color="Maroon" title="c18">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">f2</font> &amp; <font color="Maroon" title="c16">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="c14">P1</font> &amp; <font color="Maroon" title="c17">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="c14">P1</font> )
 
<b>by </b><i><a class="txt" href="#E1:73_4"><i><font color="Green" title="E72">A32</font></i></a>, <a class="txt" href="#E3:73_4"><i><font color="Green" title="E73">A33</font></i></a>, <a class="txt" href="#E9:73_4"><i><font color="Green" title="E77">A37</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E12:73_4"/>
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x2</font>
 <b>by </b><i><a class="txt" href="#E22:73"><i><font color="Green" title="E68">A23</font></i></a>, <a class="txt" href="#E5:73_4"><i><font color="Green" title="E74">A34</font></i></a>, <a class="txt" href="#E7:73_4"><i><font color="Green" title="E75">A35</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E29:73"/><b>then </b><i><font color="Green" title="E72">A39</font></i>: 
<font color="Maroon" title="c15">P1x</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<b>set </b><font color="Maroon" title="c16">I</font> =  <a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span>;<br/>
<a NAME="E30:73"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="#E1:73"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="#E16:73"><i><font color="Green" title="E64">A19</font></i></a>, <a class="ref" href="card_fin.html#T57" target="_self" title="CARD_FIN:th.57">Th57</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">XFS1</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E32:73"/><i><font color="Green" title="E73">A40</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c17">XFS1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">P1x</font>
 <b>and </b><br/><a NAME="E33:73"/><i><font color="Green" title="E74">A41</font></i>: 
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c17">XFS1</font> &amp; <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">P1x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> holds <br/><font color="Maroon" title="c17">XFS1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <span class="p2">(<span class="default"><a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b2">f</font>,1</span>)</span>
 <b>and </b><br/><a NAME="E34:73"/><i><font color="Green" title="E75">A42</font></i>: 
 <a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <span class="p1">(<span class="default"><a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c17">XFS1</font>
 <b>by </b><i><a class="txt" href="#E27:73"><i><font color="Green" title="E71">A31</font></i></a>, <a class="txt" href="#E29:73"><i><font color="Green" title="E72">A39</font></i></a>, <a class="ref" href="card_fin.html#D4" target="_self" title="CARD_FIN:def.4">Def4</a></i>;<br/>
<a NAME="E35:73"/>
( ( <font color="Maroon" title="c9">X0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">X0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; ( <font color="Maroon" title="c10">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )
 
;<br/>
<a NAME="E36:73"/><b>then </b><i><font color="Green" title="E76">A43</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">P0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">X0</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">X1</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 reconsider </b><font color="Maroon" title="c18">XP0</font> = <font color="Maroon" title="c11">P0</font>, <font color="Maroon" title="c19">XP1</font> = <font color="Maroon" title="c14">P1</font> as   <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a> <b>by </b><i><a class="ref" href="afinsq_1.html#T7" title="AFINSQ_1:th.7">AFINSQ_1:7</a></i>;<br/>
<b>set </b><font color="Maroon" title="c20">Ch</font> =  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<a NAME="E38:73"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> &amp; not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">P0</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">P1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">X0</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">X1</font> )
 
<b>by </b><i><a class="txt" href="#E1:73"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="xboole_1.html#T13" title="XBOOLE_1:th.13">XBOOLE_1:13</a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a>, <a class="ref" href="zfmisc_1.html#T140" title="ZFMISC_1:th.140">ZFMISC_1:140</a></i>;<br/>
<a NAME="E39:73"/><b>then </b><i><font color="Green" title="E77">A44</font></i>: 
( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">X0</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">X1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">X0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c18">XP0</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c19">XP1</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">XP0</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c19">XP1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">X0</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">X1</font> &amp; <font color="Maroon" title="c9">X0</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="#E12:73"><i><font color="Green" title="E62">A10</font></i></a>, <a class="txt" href="#E27:73"><i><font color="Green" title="E71">A31</font></i></a>, <a class="txt" href="#E36:73"><i><font color="Green" title="E76">A43</font></i></a>, <a class="txt" href="card_fin.html#E18"><i><font color="Green" title="E16">Lm1</font></i></a>, <a class="ref" href="card_fin.html#T17" target="_self" title="CARD_FIN:th.17">Th17</a>, <a class="ref" href="afinsq_1.html#T29" title="AFINSQ_1:th.29">AFINSQ_1:29</a></i>;<br/>
<a NAME="E40:73"/><b>then </b>
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">XP0</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c19">XP1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">XP0</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c19">XP1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c21">XP01</font> = <font color="Maroon" title="c18">XP0</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c19">XP1</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>, <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <b>by </b><i><a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/>
<a NAME="E42:73"/>
not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/>
<a NAME="E43:73"/><b>then </b>
<font color="Maroon" title="c9">X0</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c10">X1</font>
 
<b>by </b><i><a class="txt" href="card_fin.html#E18"><i><font color="Green" title="E16">Lm1</font></i></a></i>;<br/>
<a NAME="E44:73"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">P0</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">P1</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T64" title="XBOOLE_1:th.64">XBOOLE_1:64</a></i>;<br/>
<a NAME="E45:73"/><b>then </b>
<font color="Maroon" title="c21">XP01</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="#E7:73"><i><font color="Green" title="E59">A3</font></i></a>, <a class="txt" href="#E22:73"><i><font color="Green" title="E68">A23</font></i></a>, <a class="ref" href="card_fin.html#T60" target="_self" title="CARD_FIN:th.60">Th60</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c22">XFS</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E47:73"/><i><font color="Green" title="E78">A45</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c22">XFS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c21">XP01</font>
 <b>and </b><br/><a NAME="E48:73"/><i><font color="Green" title="E79">A46</font></i>: 
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c22">XFS</font> &amp; <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">XP01</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> holds <br/><font color="Maroon" title="c22">XFS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <font color="Maroon" title="c2">Fy</font>,<font color="Olive" title="b2">f</font>,1</span>)</span>
 <b>and </b><br/><a NAME="E49:73"/><i><font color="Green" title="E80">A47</font></i>: 
 <a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <font color="Maroon" title="c2">Fy</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c22">XFS</font>
 <b>by </b><i><a class="txt" href="#E1:73"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="card_fin.html#D4" target="_self" title="CARD_FIN:def.4">Def4</a></i>;<br/>
<a NAME="E50:73"/>
<font color="Maroon" title="c22">XFS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">XFS0</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c17">XFS1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_5"><b>proof </b></a><div class="add">

<a NAME="E1:73_5"/>
( (  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; (  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c1">k</font>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c1">k</font>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; ( <font color="Maroon" title="c9">X0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">X0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; ( <font color="Maroon" title="c10">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )
 
;<br/>
<a NAME="E2:73_5"/><b>then </b><i><font color="Green" title="E81">A48</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">P0x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">X0</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">P0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">X0</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">P1x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">X1</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">X1</font> )
 
<b>by </b><i><a class="txt" href="#E12:73"><i><font color="Green" title="E62">A10</font></i></a>, <a class="txt" href="#E27:73"><i><font color="Green" title="E71">A31</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:73_5"/><b>then </b>
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">XP0</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c19">XP1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c13">XFS0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c17">XFS1</font></span>)</span> &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c13">XFS0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c13">XFS0</font> )
 
<b>by </b><i><a class="txt" href="#E18:73"><i><font color="Green" title="E65">A20</font></i></a>, <a class="txt" href="#E32:73"><i><font color="Green" title="E73">A40</font></i></a>, <a class="txt" href="#E39:73"><i><font color="Green" title="E77">A44</font></i></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></i>;<br/>
<a NAME="E4:73_5"/><b>then </b><i><font color="Green" title="E82">A49</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c22">XFS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c13">XFS0</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c17">XFS1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E47:73"><i><font color="Green" title="E78">A45</font></i></a></i>;<br/>
<a NAME="E5:73_5"/><i><font color="Green" title="E83">A50</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c13">XFS0</font> holds <br/><font color="Maroon" title="c22">XFS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">XFS0</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c13">XFS0</font> implies <font color="Maroon" title="c22">XFS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">XFS0</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">n</font> )</span><br/>

<b>assume </b><a NAME="E1:73_5_1"/><i><font color="Green" title="E84">A51</font></i>: 
<font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c13">XFS0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c22">XFS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">XFS0</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">n</font></span><br/>

<b>consider </b><font color="Maroon" title="c24">fx</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:73_5_1"/><i><font color="Green" title="E85">A52</font></i>: 
( <font color="Maroon" title="c24">fx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">P0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c24">fx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X0</font> &amp; <font color="Maroon" title="c12">P0x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">fx</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E10:73"><i><font color="Green" title="E61">A9</font></i></a>, <a class="txt" href="#E18:73"><i><font color="Green" title="E65">A20</font></i></a>, <a class="txt" href="#E1:73_5_1"><i><font color="Green" title="E84">A51</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c25">fx'</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span><b> such that </b><br/><a NAME="E5:73_5_1"/><i><font color="Green" title="E86">A53</font></i>: 
( <font color="Maroon" title="c24">fx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">fx'</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">fx'</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; <font color="Maroon" title="c25">fx'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">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="#E3:73_5_1"><i><font color="Green" title="E85">A52</font></i></a></i>;<br/>
<a NAME="E6:73_5_1"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c24">fx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c24">fx</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">fx</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">fx</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="#E1:73"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="#E5:73_5_1"><i><font color="Green" title="E86">A53</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relat_1.html#T87" title="RELAT_1:th.87">RELAT_1:87</a>, <a class="ref" href="relat_1.html#T167" title="RELAT_1:th.167">RELAT_1:167</a>, <a class="ref" href="zfmisc_1.html#T140" title="ZFMISC_1:th.140">ZFMISC_1:140</a></i>;<br/>
<a NAME="E7:73_5_1"/><b>then </b><i><font color="Green" title="E87">A54</font></i>: 
( <font color="Maroon" title="c24">fx</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">fx</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c24">fx</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="#E5:73_5_1"><i><font color="Green" title="E86">A53</font></i></a>, <a class="ref" href="card_fin.html#T15" target="_self" title="CARD_FIN:th.15">Th15</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c26">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:73_5_1"/><i><font color="Green" title="E88">A55</font></i>: 
<font color="Maroon" title="c26">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">fx</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="#E5:73_5_1"><i><font color="Green" title="E86">A53</font></i></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E10:73_5_1"/>
( <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c13">XFS0</font> &amp; <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c13">XFS0</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c13">XFS0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c17">XFS1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E1:73_5_1"><i><font color="Green" title="E84">A51</font></i></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E11:73_5_1"/><b>then </b>
( <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c13">XFS0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c17">XFS1</font></span>)</span> &amp; <font color="Maroon" title="c24">fx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="txt" href="#E9:73_5_1"><i><font color="Green" title="E88">A55</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:73_5_1"/><b>then </b>
( <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c22">XFS</font> &amp; <font color="Maroon" title="c21">XP01</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">XP0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c24">fx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c26">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="c24">fx</font> )
 
<b>by </b><i><a class="txt" href="#E18:73"><i><font color="Green" title="E65">A20</font></i></a>, <a class="txt" href="#E32:73"><i><font color="Green" title="E73">A40</font></i></a>, <a class="txt" href="#E39:73"><i><font color="Green" title="E77">A44</font></i></a>, <a class="txt" href="#E47:73"><i><font color="Green" title="E78">A45</font></i></a>, <a class="txt" href="#E2:73_5"><i><font color="Green" title="E81">A48</font></i></a>, <a class="txt" href="#E1:73_5_1"><i><font color="Green" title="E84">A51</font></i></a>, <a class="txt" href="#E9:73_5_1"><i><font color="Green" title="E88">A55</font></i></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E13:73_5_1"/><b>then </b>
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c22">XFS</font> &amp; <font color="Maroon" title="c21">XP01</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">fx</font> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">fx</font> )
 
<b>by </b><i><a class="txt" href="#E3:73_5_1"><i><font color="Green" title="E85">A52</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></i>;<br/>
<a NAME="E14:73_5_1"/><b>then </b>
( <font color="Maroon" title="c22">XFS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <font color="Maroon" title="c2">Fy</font>,<font color="Maroon" title="c24">fx</font>,1</span>)</span> &amp; <font color="Maroon" title="c13">XFS0</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">fx</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,1</span>)</span> &amp;  <a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">fx</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c24">fx</font>,1 &amp;  <a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c24">fx</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <font color="Maroon" title="c2">Fy</font>,<font color="Maroon" title="c24">fx</font>,1 )
 
<b>by </b><i><a class="txt" href="#E1:73"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="#E19:73"><i><font color="Green" title="E66">A21</font></i></a>, <a class="txt" href="#E48:73"><i><font color="Green" title="E79">A46</font></i></a>, <a class="txt" href="#E1:73_5_1"><i><font color="Green" title="E84">A51</font></i></a>, <a class="txt" href="#E3:73_5_1"><i><font color="Green" title="E85">A52</font></i></a>, <a class="txt" href="#E7:73_5_1"><i><font color="Green" title="E87">A54</font></i></a>, <a class="ref" href="card_fin.html#T30" target="_self" title="CARD_FIN:th.30">Th30</a>, <a class="ref" href="card_fin.html#T32" target="_self" title="CARD_FIN:th.32">Th32</a></i>;<br/>
<b>hence </b><a NAME="E15:73_5_1"/>
<font color="Maroon" title="c22">XFS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">XFS0</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E6:73_5"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c17">XFS1</font> holds <br/><font color="Maroon" title="c22">XFS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c13">XFS0</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">XFS1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c17">XFS1</font> implies <font color="Maroon" title="c22">XFS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c13">XFS0</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">XFS1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">n</font> )</span><br/>

<b>assume </b><a NAME="E1:73_5_2"/><i><font color="Green" title="E84">A56</font></i>: 
<font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c17">XFS1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c22">XFS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c13">XFS0</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">XFS1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">n</font></span><br/>

<b>consider </b><font color="Maroon" title="c24">fx</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:73_5_2"/><i><font color="Green" title="E85">A57</font></i>: 
( <font color="Maroon" title="c24">fx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c24">fx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">X1</font> &amp; <font color="Maroon" title="c15">P1x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">fx</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E25:73"><i><font color="Green" title="E70">A30</font></i></a>, <a class="txt" href="#E32:73"><i><font color="Green" title="E73">A40</font></i></a>, <a class="txt" href="#E1:73_5_2"><i><font color="Green" title="E84">A56</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c25">fx'</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span><b> such that </b><br/><a NAME="E5:73_5_2"/><i><font color="Green" title="E86">A58</font></i>: 
( <font color="Maroon" title="c24">fx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">fx'</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">fx'</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; <font color="Maroon" title="c25">fx'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="#E3:73_5_2"><i><font color="Green" title="E85">A57</font></i></a></i>;<br/>
<a NAME="E6:73_5_2"/><i><font color="Green" title="E87">A59</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">fx</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c24">fx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_5_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:73_5_2_1"/><i><font color="Green" title="E88">A60</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c25">fx'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> )
 
<b>by </b><i><a class="txt" href="#E1:73"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="zfmisc_1.html#T140" title="ZFMISC_1:th.140">ZFMISC_1:140</a></i>;<br/>
<a NAME="E2:73_5_2_1"/><b>then </b>
( <font color="Maroon" title="c4">x</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="c24">fx</font> &amp; <font color="Maroon" title="c24">fx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="txt" href="#E1:73"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="#E5:73_5_2"><i><font color="Green" title="E86">A58</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:73_5_2_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">fx</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c24">fx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> )
 <b>by </b><i><a class="txt" href="#E5:73_5_2"><i><font color="Green" title="E86">A58</font></i></a>, <a class="txt" href="#E1:73_5_2_1"><i><font color="Green" title="E88">A60</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 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>
<a NAME="E7:73_5_2"/><i><font color="Green" title="E88">A61</font></i>: 
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">fx</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c24">fx</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_5_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:73_5_2_2"/><i><font color="Green" title="E89">A62</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c24">fx</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">fx</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">fx</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c24">fx</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c24">fx</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a>, <a class="ref" href="relat_1.html#T167" title="RELAT_1:th.167">RELAT_1:167</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c26">fx1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c24">fx</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as  <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/>
<a NAME="E3:73_5_2_2"/>
not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/>
<a NAME="E4:73_5_2_2"/><b>then </b>
not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c24">fx</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E5:73_5_2_2"/><b>then </b>
not <font color="Maroon" title="c4">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c24">fx</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E6:73_5_2_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c26">fx1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">fx</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">fx1</font> )
 
<b>by </b><i><a class="txt" href="#E1:73"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="#E5:73_5_2"><i><font color="Green" title="E86">A58</font></i></a>, <a class="txt" href="#E6:73_5_2"><i><font color="Green" title="E87">A59</font></i></a>, <a class="ref" href="card_fin.html#T13" target="_self" title="CARD_FIN:th.13">Th13</a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E7:73_5_2_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c26">fx1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c26">fx1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c26">fx1</font> )
 
<b>by </b><i><a class="txt" href="#E5:73_5_2"><i><font color="Green" title="E86">A58</font></i></a>, <a class="ref" href="card_2.html#T54" title="CARD_2:th.54">CARD_2:54</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c27">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:73_5_2_2"/><i><font color="Green" title="E90">A63</font></i>: 
<font color="Maroon" title="c27">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">fx1</font>
 <b>by </b><i><a class="txt" href="#E2:73"><i><font color="Green" title="E58">A2</font></i></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E10:73_5_2_2"/>
( <font color="Maroon" title="c27">y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c24">fx</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c24">fx</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="txt" href="#E9:73_5_2_2"><i><font color="Green" title="E90">A63</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E11:73_5_2_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c24">fx</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">fx</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c24">fx</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E12:73_5_2_2"/>
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">fx</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c24">fx</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="#E1:73_5_2_2"><i><font color="Green" title="E89">A62</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E8:73_5_2"/>
( <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c17">XFS1</font> &amp; <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c19">XP1</font> )
 
<b>by </b><i><a class="txt" href="#E32:73"><i><font color="Green" title="E73">A40</font></i></a>, <a class="txt" href="#E2:73_5"><i><font color="Green" title="E81">A48</font></i></a>, <a class="txt" href="#E1:73_5_2"><i><font color="Green" title="E84">A56</font></i></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></i>;<br/>
<a NAME="E9:73_5_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c13">XFS0</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c13">XFS0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c17">XFS1</font></span>)</span> &amp; <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c19">XP1</font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E10:73_5_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c13">XFS0</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c22">XFS</font> &amp; <font color="Maroon" title="c19">XP1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">XP01</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c18">XP0</font></span>)</span></span>)</span> &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c13">XFS0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c18">XP0</font> )
 
<b>by </b><i><a class="txt" href="#E18:73"><i><font color="Green" title="E65">A20</font></i></a>, <a class="txt" href="#E32:73"><i><font color="Green" title="E73">A40</font></i></a>, <a class="txt" href="#E39:73"><i><font color="Green" title="E77">A44</font></i></a>, <a class="txt" href="#E47:73"><i><font color="Green" title="E78">A45</font></i></a>, <a class="txt" href="#E2:73_5"><i><font color="Green" title="E81">A48</font></i></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></i>;<br/>
<a NAME="E11:73_5_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c13">XFS0</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c22">XFS</font> &amp; <font color="Maroon" title="c21">XP01</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c18">XP0</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">fx</font> &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c18">XP0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c13">XFS0</font> )
 
<b>by </b><i><a class="txt" href="#E3:73_5_2"><i><font color="Green" title="E85">A57</font></i></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></i>;<br/>
<a NAME="E12:73_5_2"/><b>then </b>
( <font color="Maroon" title="c22">XFS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c13">XFS0</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <font color="Maroon" title="c2">Fy</font>,<font color="Maroon" title="c24">fx</font>,1</span>)</span> &amp; <font color="Maroon" title="c17">XFS1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <span class="p2">(<span class="default"><a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">fx</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,1</span>)</span> &amp;  <a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <span class="p1">(<span class="default"><a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">fx</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">fx</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,1</span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span> &amp;  <a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">fx</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <font color="Maroon" title="c2">Fy</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">fx</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,1 )
 
<b>by </b><i><a class="txt" href="#E1:73"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="#E16:73"><i><font color="Green" title="E64">A19</font></i></a>, <a class="txt" href="#E33:73"><i><font color="Green" title="E74">A41</font></i></a>, <a class="txt" href="#E48:73"><i><font color="Green" title="E79">A46</font></i></a>, <a class="txt" href="#E1:73_5_2"><i><font color="Green" title="E84">A56</font></i></a>, <a class="txt" href="#E3:73_5_2"><i><font color="Green" title="E85">A57</font></i></a>, <a class="txt" href="#E7:73_5_2"><i><font color="Green" title="E88">A61</font></i></a>, <a class="ref" href="card_fin.html#T32" target="_self" title="CARD_FIN:th.32">Th32</a>, <a class="ref" href="card_fin.html#T59" target="_self" title="CARD_FIN:th.59">Th59</a></i>;<br/>
<a NAME="E13:73_5_2"/><b>then </b>
( <font color="Maroon" title="c22">XFS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c13">XFS0</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <font color="Maroon" title="c2">Fy</font>,<font color="Maroon" title="c24">fx</font>,1</span>)</span> &amp; <font color="Maroon" title="c17">XFS1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <font color="Maroon" title="c2">Fy</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c24">fx</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,1</span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <font color="Maroon" title="c2">Fy</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">fx</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,1</span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <font color="Maroon" title="c2">Fy</font>,<font color="Maroon" title="c24">fx</font>,1 )
 
<b>by </b><i><a class="txt" href="#E6:73_5_2"><i><font color="Green" title="E87">A59</font></i></a>, <a class="ref" href="card_fin.html#T34" target="_self" title="CARD_FIN:th.34">Th34</a></i>;<br/>
<b>hence </b><a NAME="E14:73_5_2"/>
<font color="Maroon" title="c22">XFS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c13">XFS0</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">XFS1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:73_5"/>
<font color="Maroon" title="c22">XFS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">XFS0</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c17">XFS1</font>
 <b>by </b><i><a class="txt" href="#E4:73_5"><i><font color="Green" title="E82">A49</font></i></a>, <a class="txt" href="#E5:73_5"><i><font color="Green" title="E83">A50</font></i></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</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="E51:73"/><b>then </b>
( <a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c22">XFS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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"><a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c13">XFS0</font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c17">XFS1</font></span>)</span> &amp; <a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c13">XFS0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> &amp; <a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c17">XFS1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <span class="p1">(<span class="default"><a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">k</font> &amp; <a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c22">XFS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <font color="Maroon" title="c2">Fy</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="#E20:73"><i><font color="Green" title="E67">A22</font></i></a>, <a class="txt" href="#E34:73"><i><font color="Green" title="E75">A42</font></i></a>, <a class="txt" href="#E49:73"><i><font color="Green" title="E80">A47</font></i></a>, <a class="ref" href="stirl2_1.html#T47" title="STIRL2_1:th.47">STIRL2_1:47</a>, <a class="ref" href="stirl2_1.html#D4" title="STIRL2_1:def.4">STIRL2_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E52:73"/>
 <a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <font color="Maroon" title="c2">Fy</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <span class="p2">(<span class="default"><a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="binop_2.html#D23" title="BINOP_2:def.23">BINOP_2:def 23</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>
