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<span class="kw">let </span><font color="Maroon" title="c1">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font> being  <a href="membered.html#V6" title="MEMBERED:attr.6">natural-membered</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="afinsq_2.html#R1" title="AFINSQ_2:pred.1">&lt;N&lt;</a> <font color="Olive" title="b2">Y</font> &amp; <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Olive" title="b2">Y</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Olive" title="b2">Y</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</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> <span class="p3">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Olive" title="b1">X</font></span>)</span></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font>, <font color="Maroon" title="c3">Y</font> be   <a href="membered.html#V6" title="MEMBERED:attr.6">natural-membered</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">X</font> <a href="afinsq_2.html#R1" title="AFINSQ_2:pred.1">&lt;N&lt;</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c3">Y</font></span>)</span> implies <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</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> <span class="p3">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span></span>)</span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:62"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
<font color="Maroon" title="c2">X</font> <a href="afinsq_2.html#R1" title="AFINSQ_2:pred.1">&lt;N&lt;</a> <font color="Maroon" title="c3">Y</font>
 <span class="kw">and </span><br/><a NAME="E2:62"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c3">Y</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</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> <span class="p3">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:62"/><span class="lab"><font color="Green" title="E38">A3</font></span>: 
<font color="Maroon" title="c3">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T2" target="_self" title="AFINSQ_2:th.2">Th2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">h</font> = <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> as   <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>  ;<br/>
<a NAME="E6:62"/><span class="lab"><font color="Green" title="E39">A4</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c2">X</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> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T31" target="_self" title="AFINSQ_2:th.31">Th31</a></span>;<br/>
<a NAME="E7:62"/><span class="lab"><font color="Green" title="E40">A5</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c3">Y</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> <font color="Maroon" title="c3">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T31" target="_self" title="AFINSQ_2:th.31">Th31</a></span>;<br/>
<a NAME="E8:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A6</font></span>: 
<font color="Maroon" title="c5">h</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E6:62"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="ref" href="afinsq_2.html#T41" target="_self" title="AFINSQ_2:th.41">Th41</a></span>;<br/>
<a NAME="E9:62"/><span class="lab"><font color="Green" title="E42">A7</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></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"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T31" target="_self" title="AFINSQ_2:th.31">Th31</a></span>;<br/>
<a NAME="E10:62"/>
<font color="Maroon" title="c4">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="membered.html#T6" title="MEMBERED:th.6">MEMBERED:6</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">Y</font> = 
<font color="Maroon" title="c2">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:62"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></span>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="afinsq_2.html#T34" target="_self" title="AFINSQ_2:th.34">Th34</a></span>
;<br/>
<a NAME="E12:62"/><span class="kw">then </span>
<font color="Maroon" title="c2">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<a NAME="E13:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A8</font></span>: 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">Y</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="c2">X</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"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T53" title="CARD_2:th.53">CARD_2:53</a></span>;<br/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">h</font> = 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#D2" target="_self" title="AFINSQ_2:def.2">Def2</a></span>
<br/>.= 
<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="c2">X</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:62"><span class="lab"><font color="Green" title="E43">A8</font></span></a>, <a class="txt" href="#E9:62"><span class="lab"><font color="Green" title="E42">A7</font></span></a>, <a class="ref" href="afinsq_2.html#T31" target="_self" title="AFINSQ_2:th.31">Th31</a></span>
<br/>.= 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">Y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>
<br/>.= 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c3">Y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T31" target="_self" title="AFINSQ_2:th.31">Th31</a></span>
;<br/>
<a NAME="E15:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A9</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">h</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="c3">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T31" target="_self" title="AFINSQ_2:th.31">Th31</a></span>;<br/>
<a NAME="E16:62"/><span class="lab"><font color="Green" title="E45">A10</font></span>: 
for <font color="Olive" title="b1">l</font>, <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k1</font>, <font color="Olive" title="b4">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">h</font> &amp; <font color="Olive" title="b3">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">l</font> &amp; <font color="Olive" title="b4">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b3">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">k2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">l</font>, <font color="Maroon" title="c7">m</font>, <font color="Maroon" title="c8">k1</font>, <font color="Maroon" title="c9">k2</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">m</font> &amp; <font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">h</font> &amp; <font color="Maroon" title="c8">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">l</font> &amp; <font color="Maroon" title="c9">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">m</font> implies <font color="Maroon" title="c8">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:62_3"/><span class="lab"><font color="Green" title="E46">A11</font></span>: 
<font color="Maroon" title="c6">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">m</font>
 <span class="kw">and </span><br/><a NAME="E2:62_3"/><span class="lab"><font color="Green" title="E47">A12</font></span>: 
<font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">h</font>
 <span class="kw">and </span><br/><a NAME="E3:62_3"/><span class="lab"><font color="Green" title="E48">A13</font></span>: 
<font color="Maroon" title="c8">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">l</font>
 <span class="kw">and </span><br/><a NAME="E4:62_3"/><span class="lab"><font color="Green" title="E49">A14</font></span>: 
<font color="Maroon" title="c9">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k2</font></span><br/>

<a NAME="E5:62_3"/><span class="lab"><font color="Green" title="E50">A15</font></span>: 
<font color="Maroon" title="c7">m</font> <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="c2">X</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:62"><span class="lab"><font color="Green" title="E43">A8</font></span></a>, <a class="txt" href="#E9:62"><span class="lab"><font color="Green" title="E42">A7</font></span></a>, <a class="txt" href="#E15:62"><span class="lab"><font color="Green" title="E44">A9</font></span></a>, <a class="txt" href="#E2:62_3"><span class="lab"><font color="Green" title="E47">A12</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">m3</font> = <font color="Maroon" title="c7">m</font> <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="c2">X</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">l3</font> = <font color="Maroon" title="c6">l</font> <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="c2">X</font></span>)</span>;<br/>
<a NAME="E6:62_3"/><span class="lab"><font color="Green" title="E51">A16</font></span>: 
<font color="Maroon" title="c6">l</font> <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="c2">X</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">m</font> <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="c2">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_3"><span class="lab"><font color="Green" title="E46">A11</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E7:62_3"/>
<font color="Maroon" title="c6">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:62"><span class="lab"><font color="Green" title="E44">A9</font></span></a>, <a class="txt" href="#E1:62_3"><span class="lab"><font color="Green" title="E46">A11</font></span></a>, <a class="txt" href="#E2:62_3"><span class="lab"><font color="Green" title="E47">A12</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E8:62_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A17</font></span>: 
<font color="Maroon" title="c5">h</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E6:62"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="ref" href="afinsq_2.html#T41" target="_self" title="AFINSQ_2:th.41">Th41</a></span>;<br/>
<a NAME="E9:62_3"/>
<font color="Maroon" title="c5">h</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E6:62"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="txt" href="#E15:62"><span class="lab"><font color="Green" title="E44">A9</font></span></a>, <a class="txt" href="#E2:62_3"><span class="lab"><font color="Green" title="E47">A12</font></span></a>, <a class="ref" href="afinsq_2.html#T41" target="_self" title="AFINSQ_2:th.41">Th41</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:62_3"/>
<font color="Maroon" title="c8">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_3"><span class="lab"><font color="Green" title="E48">A13</font></span></a>, <a class="txt" href="#E4:62_3"><span class="lab"><font color="Green" title="E49">A14</font></span></a>, <a class="txt" href="#E8:62_3"><span class="lab"><font color="Green" title="E52">A17</font></span></a>, <a class="txt" href="#E5:62_3"><span class="lab"><font color="Green" title="E50">A15</font></span></a>, <a class="txt" href="#E6:62_3"><span class="lab"><font color="Green" title="E51">A16</font></span></a>, <a class="ref" href="afinsq_2.html#D5" target="_self" title="AFINSQ_2:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E17:62"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E6:62"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="txt" href="#E7:62"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="ref" href="afinsq_2.html#T41" target="_self" title="AFINSQ_2:th.41">Th41</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:62"/>
<span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</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> <span class="p3">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:62"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="txt" href="#E16:62"><span class="lab"><font color="Green" title="E45">A10</font></span></a>, <a class="txt" href="#E8:62"><span class="lab"><font color="Green" title="E41">A6</font></span></a>, <a class="ref" href="afinsq_2.html#D5" target="_self" title="AFINSQ_2:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
