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<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>, <font color="Olive" title="b3">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b4">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="b5">P</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> <span class="p1">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Olive" title="b4">X</font>,<font color="Maroon" title="c1">k</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font></span>)</span>, <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Olive" title="b4">X</font>,<font color="Maroon" title="c1">k</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>  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="b4">X</font> &amp; <font color="Olive" title="b5">P</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">y</font> holds <br/>for <font color="Olive" title="b6">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>   st  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Olive" title="b6">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="Olive" title="b5">P</font> &amp; ( for <font color="Olive" title="b7">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b8">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b7">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="Olive" title="b6">XFS</font> &amp; <font color="Olive" title="b8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">z</font> holds <br/><font color="Olive" title="b6">XFS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b7">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="Olive" title="b1">Fy</font>,<font color="Olive" title="b8">f</font>,<font color="Olive" title="b2">x</font></span>)</span> ) holds <br/> <a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <font color="Olive" title="b1">Fy</font>,<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="Olive" title="b6">XFS</font></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>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b3">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="b4">P</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> <span class="p1">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Olive" title="b3">X</font>,<font color="Maroon" title="c1">k</font>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span>)</span>, <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Olive" title="b3">X</font>,<font color="Maroon" title="c1">k</font>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>  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="b3">X</font> &amp; <font color="Olive" title="b4">P</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> holds <br/>for <font color="Olive" title="b5">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>   st  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Olive" title="b5">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="Olive" title="b4">P</font> &amp; ( for <font color="Olive" title="b6">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b7">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b6">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="Olive" title="b5">XFS</font> &amp; <font color="Olive" title="b7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">z</font> holds <br/><font color="Olive" title="b5">XFS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b6">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="b7">f</font>,<font color="Olive" title="b1">x</font></span>)</span> ) holds <br/> <a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <font color="Maroon" title="c2">Fy</font>,<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="Olive" title="b5">XFS</font></span><br/><b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</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">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">P</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> <span class="p1">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>, <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>  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">P</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font> holds <br/>for <font color="Olive" title="b3">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>   st  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Olive" title="b3">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="Olive" title="b2">P</font> &amp; ( for <font color="Olive" title="b4">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b5">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b4">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="Olive" title="b3">XFS</font> &amp; <font color="Olive" title="b5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">z</font> holds <br/><font color="Olive" title="b3">XFS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b4">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="b5">f</font>,<font color="Maroon" title="c3">x</font></span>)</span> ) holds <br/> <a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <font color="Maroon" title="c2">Fy</font>,<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="Olive" title="b3">XFS</font></span><br/><b>let </b><font color="Maroon" title="c5">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">P</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> <span class="p1">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>, <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>  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="c5">X</font> &amp; <font color="Olive" title="b1">P</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font> holds <br/>for <font color="Olive" title="b2">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>   st  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Olive" title="b2">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="Olive" title="b1">P</font> &amp; ( for <font color="Olive" title="b3">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b4">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b3">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="Olive" title="b2">XFS</font> &amp; <font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">z</font> holds <br/><font color="Olive" title="b2">XFS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b3">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="b4">f</font>,<font color="Maroon" title="c3">x</font></span>)</span> ) holds <br/> <a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <font color="Maroon" title="c2">Fy</font>,<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="Olive" title="b2">XFS</font></span><br/><b>let </b><font color="Maroon" title="c6">P</font> be   <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="c5">X</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>, <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>; <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="c5">X</font> &amp; <font color="Maroon" title="c6">P</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font> implies for <font color="Olive" title="b1">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>   st  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Olive" title="b1">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="c6">P</font> &amp; ( for <font color="Olive" title="b2">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b3">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b2">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="Olive" title="b1">XFS</font> &amp; <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">z</font> holds <br/><font color="Olive" title="b1">XFS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">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="b3">f</font>,<font color="Maroon" title="c3">x</font></span>)</span> ) holds <br/> <a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <font color="Maroon" title="c2">Fy</font>,<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="Olive" title="b1">XFS</font> )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:61"/><i><font color="Green" title="E47">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="c5">X</font>
 <b>and </b><br/><a NAME="E2:61"/><i><font color="Green" title="E48">A2</font></i>: 
<font color="Maroon" title="c6">P</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>and </b><br/><a NAME="E3:61"/><i><font color="Green" title="E49">A3</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">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>   st  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Olive" title="b1">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="c6">P</font> &amp; ( for <font color="Olive" title="b2">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b3">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b2">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="Olive" title="b1">XFS</font> &amp; <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">z</font> holds <br/><font color="Olive" title="b1">XFS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">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="b3">f</font>,<font color="Maroon" title="c3">x</font></span>)</span> ) holds <br/> <a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <font color="Maroon" title="c2">Fy</font>,<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="Olive" title="b1">XFS</font></span><br/>

<b>let </b><font color="Maroon" title="c7">XFS1</font> be   <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">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="c6">P</font> &amp; ( 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="c7">XFS1</font> &amp; <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P</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="c7">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> <font color="Maroon" title="c2">Fy</font>,<font color="Olive" title="b2">f</font>,<font color="Maroon" title="c3">x</font></span>)</span> ) implies  <a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <font color="Maroon" title="c2">Fy</font>,<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="c7">XFS1</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E4:61"/><i><font color="Green" title="E50">A4</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">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="c6">P</font>
 <b>and </b><br/><a NAME="E5:61"/><i><font color="Green" title="E51">A5</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="c7">XFS1</font> &amp; <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P</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="c7">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> <font color="Maroon" title="c2">Fy</font>,<font color="Olive" title="b2">f</font>,<font color="Maroon" title="c3">x</font></span>)</span>
 ; <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>,<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="c7">XFS1</font></span><br/>

<b>consider </b><font color="Maroon" title="c8">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="E7:61"/><i><font color="Green" title="E52">A6</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">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="c6">P</font>
 <b>and </b><br/><a NAME="E8:61"/><i><font color="Green" title="E53">A7</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="c8">XFS</font> &amp; <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P</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="c8">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>,<font color="Maroon" title="c3">x</font></span>)</span>
 <b>and </b><br/><a NAME="E9:61"/><i><font color="Green" title="E54">A8</font></i>: 
 <a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <font color="Maroon" title="c2">Fy</font>,<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="c8">XFS</font>
 <b>by </b><i><a class="txt" href="#E1:61"><i><font color="Green" title="E47">A1</font></i></a>, <a class="txt" href="#E2:61"><i><font color="Green" title="E48">A2</font></i></a>, <a class="txt" href="#E3:61"><i><font color="Green" title="E49">A3</font></i></a>, <a class="ref" href="card_fin.html#D4" target="_self" title="CARD_FIN:def.4">Def4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">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="c9">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="c8">XFS</font> implies <font color="Maroon" title="c7">XFS1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">XFS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">z</font> )</span><br/><b>assume </b><a NAME="E1:61_1"/><i><font color="Green" title="E55">A9</font></i>: 
<font color="Maroon" title="c9">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="c8">XFS</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">XFS1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">XFS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">z</font></span><br/><a NAME="E2:61_1"/>
<font color="Maroon" title="c6">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">P</font>
 <b>by </b><i><a class="txt" href="#E7:61"><i><font color="Green" title="E52">A6</font></i></a>, <a class="txt" href="#E1:61_1"><i><font color="Green" title="E55">A9</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">Pz</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">X</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span><b> such that </b><br/><a NAME="E4:61_1"/><i><font color="Green" title="E56">A10</font></i>: 
( <font color="Maroon" title="c10">Pz</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z</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="c10">Pz</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">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="c1">k</font> )
 <b>by </b><i><a class="ref" href="card_fin.html#D1" target="_self" title="CARD_FIN:def.1">Def1</a></i>;<br/><a NAME="E5:61_1"/>
( <font color="Maroon" title="c7">XFS1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">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="Maroon" title="c10">Pz</font>,<font color="Maroon" title="c3">x</font></span>)</span> &amp; <font color="Maroon" title="c8">XFS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">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="Maroon" title="c10">Pz</font>,<font color="Maroon" title="c3">x</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E4:61"><i><font color="Green" title="E50">A4</font></i></a>, <a class="txt" href="#E5:61"><i><font color="Green" title="E51">A5</font></i></a>, <a class="txt" href="#E7:61"><i><font color="Green" title="E52">A6</font></i></a>, <a class="txt" href="#E8:61"><i><font color="Green" title="E53">A7</font></i></a>, <a class="txt" href="#E1:61_1"><i><font color="Green" title="E55">A9</font></i></a>, <a class="txt" href="#E4:61_1"><i><font color="Green" title="E56">A10</font></i></a></i>;<br/><b>hence </b><a NAME="E6:61_1"/>
<font color="Maroon" title="c7">XFS1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">XFS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">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>
<b>hence </b><a NAME="E11:61"/>
 <a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <font color="Maroon" title="c2">Fy</font>,<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="c7">XFS1</font>
 <b>by </b><i><a class="txt" href="#E4:61"><i><font color="Green" title="E50">A4</font></i></a>, <a class="txt" href="#E7:61"><i><font color="Green" title="E52">A6</font></i></a>, <a class="txt" href="#E9:61"><i><font color="Green" title="E54">A8</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</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>
