<?xml version="1.0"?>
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<span class="kw">let </span><font color="Maroon" title="c1">x1</font>, <font color="Maroon" title="c2">x2</font>, <font color="Maroon" title="c3">y1</font>, <font color="Maroon" title="c4">y2</font>, <font color="Maroon" title="c5">z1</font>, <font color="Maroon" title="c6">z2</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>; <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"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">z2</font> iff ( ( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x1</font> &amp; <font color="Maroon" title="c6">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x2</font> ) or ( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font> &amp; <font color="Maroon" title="c6">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x2</font> ) or ( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x1</font> &amp; <font color="Maroon" title="c6">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font> ) or ( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font> &amp; <font color="Maroon" title="c6">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font> ) ) )</span><br/>











<span class="kw">set </span><font color="Maroon" title="c7">H</font> = <font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">Hz</font> = <font color="Maroon" title="c5">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">z2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">f</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font>;<br/>
<a NAME="E1:162"/><span class="lab"><font color="Green" title="E54">A1</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T148" target="_self" title="ZF_LANG1:th.148">Th148</a>, <a class="ref" href="zf_lang.html#T31" title="ZF_LANG:th.31">ZF_LANG:31</a></span>;<br/>
<a NAME="E2:162"/>
<font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font> is  <a href="zf_lang.html#V3" title="ZF_LANG:attr.3">being_membership</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T16" title="ZF_LANG:th.16">ZF_LANG:16</a></span>;<br/>
<a NAME="E3:162"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A2</font></span>: 
<font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font> is  <a href="zf_lang.html#V7" title="ZF_LANG:attr.7">atomic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#D15" title="ZF_LANG:def.15">ZF_LANG:def 15</a></span>;<br/>
<a NAME="E4:162"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T27" title="ZF_LANG:th.27">ZF_LANG:27</a></span>;<br/>
<a NAME="E5:162"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A4</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default">1,2,3</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="finseq_3.html#T1" title="FINSEQ_3:th.1">FINSEQ_3:1</a></span>;<br/>
<a NAME="E6:162"/>
<font color="Maroon" title="c5">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">z2</font> is  <a href="zf_lang.html#V3" title="ZF_LANG:attr.3">being_membership</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T16" title="ZF_LANG:th.16">ZF_LANG:16</a></span>;<br/>
<a NAME="E7:162"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A5</font></span>: 
<font color="Maroon" title="c5">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">z2</font> is  <a href="zf_lang.html#V7" title="ZF_LANG:attr.7">atomic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#D15" title="ZF_LANG:def.15">ZF_LANG:def 15</a></span>;<br/>
<a NAME="E8:162"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">z2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T27" title="ZF_LANG:th.27">ZF_LANG:27</a></span>;<br/>
<a NAME="E9:162"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A6</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">z2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E10:162"/>
 <a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">z2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T2" target="_self" title="ZF_LANG1:th.2">Th2</a></span>;<br/>
<a NAME="E11:162"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A7</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">z2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:162"><span class="lab"><font color="Green" title="E58">A5</font></span></a>, <a class="ref" href="zf_lang.html#T52" title="ZF_LANG:th.52">ZF_LANG:52</a></span>;<br/>
<a NAME="E12:162"/>
 <a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">z2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T2" target="_self" title="ZF_LANG1:th.2">Th2</a></span>;<br/>
<a NAME="E13:162"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A8</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">z2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:162"><span class="lab"><font color="Green" title="E58">A5</font></span></a>, <a class="ref" href="zf_lang.html#T52" title="ZF_LANG:th.52">ZF_LANG:52</a></span>;<br/>
<a NAME="E14:162"/><span class="lab"><font color="Green" title="E62">A9</font></span>: 
 <a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T2" target="_self" title="ZF_LANG1:th.2">Th2</a></span>;<br/>
<a NAME="E15:162"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A10</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:162"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="ref" href="zf_lang.html#T52" title="ZF_LANG:th.52">ZF_LANG:52</a></span>;<br/>
<a NAME="E16:162"/><span class="lab"><font color="Green" title="E64">A11</font></span>: 
 <a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T2" target="_self" title="ZF_LANG1:th.2">Th2</a></span>;<br/>
<a NAME="E17:162"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A12</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:162"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="ref" href="zf_lang.html#T52" title="ZF_LANG:th.52">ZF_LANG:52</a></span>;<br/>
<span class="kw">thus </span><a NAME="E18:162"/>
(  not <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">z2</font> or ( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x1</font> &amp; <font color="Maroon" title="c6">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x2</font> ) or ( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font> &amp; <font color="Maroon" title="c6">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x2</font> ) or ( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x1</font> &amp; <font color="Maroon" title="c6">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font> ) or ( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font> &amp; <font color="Maroon" title="c6">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</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="c1">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x1</font> &amp; <font color="Maroon" title="c6">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x2</font> ) or ( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font> &amp; <font color="Maroon" title="c6">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x2</font> ) or ( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x1</font> &amp; <font color="Maroon" title="c6">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font> ) or ( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font> &amp; <font color="Maroon" title="c6">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font> ) ) implies <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">z2</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="162_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:162_1"/><span class="lab"><font color="Green" title="E66">A13</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">z2</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="c1">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x1</font> &amp; <font color="Maroon" title="c6">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x2</font> ) or ( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font> &amp; <font color="Maroon" title="c6">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x2</font> ) or ( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x1</font> &amp; <font color="Maroon" title="c6">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font> ) or ( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font> &amp; <font color="Maroon" title="c6">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font> ) )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="162_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:162_1_1"/>
( ( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y1</font> ) or ( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y1</font> ) or ( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font> ) or ( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="162_1_1_1"><span class="kw">case </span></a><a NAME="E1:162_1_1_1"/><span class="lab"><font color="Green" title="E67">A14</font></span>: 
( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y1</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="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x1</font> &amp; <font color="Maroon" title="c6">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x2</font> )</span><br/><div class="add"><a NAME="E2:162_1_1_1"/>
( 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> &amp; 3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:162"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:162_1_1_1"/>
( <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x1</font> &amp; <font color="Maroon" title="c6">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:162"><span class="lab"><font color="Green" title="E65">A12</font></span></a>, <a class="txt" href="#E15:162"><span class="lab"><font color="Green" title="E63">A10</font></span></a>, <a class="txt" href="#E11:162"><span class="lab"><font color="Green" title="E60">A7</font></span></a>, <a class="txt" href="#E13:162"><span class="lab"><font color="Green" title="E61">A8</font></span></a>, <a class="txt" href="#E1:162_1"><span class="lab"><font color="Green" title="E66">A13</font></span></a>, <a class="txt" href="#E1:162_1_1_1"><span class="lab"><font color="Green" title="E67">A14</font></span></a>, <a class="ref" href="zf_lang1.html#D4" target="_self" title="ZF_LANG1:def.4">Def4</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="162_1_1_2"><span class="kw">case </span></a><a NAME="E1:162_1_1_2"/><span class="lab"><font color="Green" title="E67">A15</font></span>: 
( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y1</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="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font> &amp; <font color="Maroon" title="c6">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x2</font> )</span><br/><div class="add"><a NAME="E2:162_1_1_2"/><span class="lab"><font color="Green" title="E68">A16</font></span>: 
( 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> &amp; 3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:162"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></span>;<br/><a NAME="E3:162_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:162"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="txt" href="#E16:162"><span class="lab"><font color="Green" title="E64">A11</font></span></a>, <a class="txt" href="#E1:162_1_1_2"><span class="lab"><font color="Green" title="E67">A15</font></span></a>, <a class="ref" href="zf_lang.html#T52" title="ZF_LANG:th.52">ZF_LANG:52</a></span>;<br/><span class="kw">hence </span><a NAME="E4:162_1_1_2"/>
( <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font> &amp; <font color="Maroon" title="c6">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:162"><span class="lab"><font color="Green" title="E63">A10</font></span></a>, <a class="txt" href="#E11:162"><span class="lab"><font color="Green" title="E60">A7</font></span></a>, <a class="txt" href="#E13:162"><span class="lab"><font color="Green" title="E61">A8</font></span></a>, <a class="txt" href="#E1:162_1"><span class="lab"><font color="Green" title="E66">A13</font></span></a>, <a class="txt" href="#E1:162_1_1_2"><span class="lab"><font color="Green" title="E67">A15</font></span></a>, <a class="txt" href="#E2:162_1_1_2"><span class="lab"><font color="Green" title="E68">A16</font></span></a>, <a class="ref" href="zf_lang1.html#D4" target="_self" title="ZF_LANG1:def.4">Def4</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="162_1_1_3"><span class="kw">case </span></a><a NAME="E1:162_1_1_3"/><span class="lab"><font color="Green" title="E67">A17</font></span>: 
( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</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="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x1</font> &amp; <font color="Maroon" title="c6">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font> )</span><br/><div class="add"><a NAME="E2:162_1_1_3"/><span class="lab"><font color="Green" title="E68">A18</font></span>: 
( 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> &amp; 3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:162"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></span>;<br/><a NAME="E3:162_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:162"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="txt" href="#E14:162"><span class="lab"><font color="Green" title="E62">A9</font></span></a>, <a class="txt" href="#E1:162_1_1_3"><span class="lab"><font color="Green" title="E67">A17</font></span></a>, <a class="ref" href="zf_lang.html#T52" title="ZF_LANG:th.52">ZF_LANG:52</a></span>;<br/><span class="kw">hence </span><a NAME="E4:162_1_1_3"/>
( <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x1</font> &amp; <font color="Maroon" title="c6">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:162"><span class="lab"><font color="Green" title="E65">A12</font></span></a>, <a class="txt" href="#E11:162"><span class="lab"><font color="Green" title="E60">A7</font></span></a>, <a class="txt" href="#E13:162"><span class="lab"><font color="Green" title="E61">A8</font></span></a>, <a class="txt" href="#E1:162_1"><span class="lab"><font color="Green" title="E66">A13</font></span></a>, <a class="txt" href="#E1:162_1_1_3"><span class="lab"><font color="Green" title="E67">A17</font></span></a>, <a class="txt" href="#E2:162_1_1_3"><span class="lab"><font color="Green" title="E68">A18</font></span></a>, <a class="ref" href="zf_lang1.html#D4" target="_self" title="ZF_LANG1:def.4">Def4</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="162_1_1_4"><span class="kw">case </span></a><a NAME="E1:162_1_1_4"/><span class="lab"><font color="Green" title="E67">A19</font></span>: 
( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</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="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font> &amp; <font color="Maroon" title="c6">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font> )</span><br/><div class="add"><a NAME="E2:162_1_1_4"/><span class="lab"><font color="Green" title="E68">A20</font></span>: 
( 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> &amp; 3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:162"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></span>;<br/><a NAME="E3:162_1_1_4"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:162"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="txt" href="#E16:162"><span class="lab"><font color="Green" title="E64">A11</font></span></a>, <a class="txt" href="#E14:162"><span class="lab"><font color="Green" title="E62">A9</font></span></a>, <a class="txt" href="#E1:162_1_1_4"><span class="lab"><font color="Green" title="E67">A19</font></span></a>, <a class="ref" href="zf_lang.html#T52" title="ZF_LANG:th.52">ZF_LANG:52</a></span>;<br/><span class="kw">hence </span><a NAME="E4:162_1_1_4"/>
( <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font> &amp; <font color="Maroon" title="c6">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:162"><span class="lab"><font color="Green" title="E60">A7</font></span></a>, <a class="txt" href="#E13:162"><span class="lab"><font color="Green" title="E61">A8</font></span></a>, <a class="txt" href="#E1:162_1"><span class="lab"><font color="Green" title="E66">A13</font></span></a>, <a class="txt" href="#E2:162_1_1_4"><span class="lab"><font color="Green" title="E68">A20</font></span></a>, <a class="ref" href="zf_lang1.html#D4" target="_self" title="ZF_LANG1:def.4">Def4</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>

<a NAME="E19:162"/><span class="lab"><font color="Green" title="E66">A21</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:162"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E20:162"/><span class="lab"><font color="Green" title="E67">A22</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#D4" target="_self" title="ZF_LANG1:def.4">Def4</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E68">A23</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="162_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:162_2"/><span class="lab"><font color="Green" title="E69">A24</font></span>: 
( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x1</font> &amp; <font color="Maroon" title="c6">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x2</font> )
 ; <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"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">z2</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="162_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">a</font> be    <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="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">z2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font> )</span><br/><span class="kw">assume </span><a NAME="E1:162_2_1"/><span class="lab"><font color="Green" title="E70">A25</font></span>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</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"><font color="Maroon" title="c5">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">z2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font></span><br/><a NAME="E2:162_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:162"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:162_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">z2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:162"><span class="lab"><font color="Green" title="E65">A12</font></span></a>, <a class="txt" href="#E15:162"><span class="lab"><font color="Green" title="E63">A10</font></span></a>, <a class="txt" href="#E1:162"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="txt" href="#E1:162_2"><span class="lab"><font color="Green" title="E69">A24</font></span></a>, <a class="txt" href="#E1:162_2_1"><span class="lab"><font color="Green" title="E70">A25</font></span></a>, <a class="ref" href="zf_lang1.html#D4" target="_self" title="ZF_LANG1:def.4">Def4</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><span class="kw">hence </span><a NAME="E3:162_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">z2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:162"><span class="lab"><font color="Green" title="E67">A22</font></span></a>, <a class="txt" href="#E19:162"><span class="lab"><font color="Green" title="E66">A21</font></span></a>, <a class="txt" href="#E9:162"><span class="lab"><font color="Green" title="E59">A6</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</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="E22:162"/><span class="lab"><font color="Green" title="E69">A26</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">z2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T31" title="ZF_LANG:th.31">ZF_LANG:31</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E70">A27</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="162_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:162_3"/><span class="lab"><font color="Green" title="E71">A28</font></span>: 
( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x1</font> &amp; <font color="Maroon" title="c6">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font> )
 ; <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"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">z2</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="162_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">a</font> be    <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="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">z2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font> )</span><br/><span class="kw">assume </span><a NAME="E1:162_3_1"/><span class="lab"><font color="Green" title="E72">A29</font></span>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</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"><font color="Maroon" title="c5">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">z2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font></span><br/><a NAME="E2:162_3_1"/><span class="kw">then </span>
( <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:162"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:162_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">z2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:162"><span class="lab"><font color="Green" title="E65">A12</font></span></a>, <a class="txt" href="#E15:162"><span class="lab"><font color="Green" title="E63">A10</font></span></a>, <a class="txt" href="#E22:162"><span class="lab"><font color="Green" title="E69">A26</font></span></a>, <a class="txt" href="#E11:162"><span class="lab"><font color="Green" title="E60">A7</font></span></a>, <a class="txt" href="#E13:162"><span class="lab"><font color="Green" title="E61">A8</font></span></a>, <a class="txt" href="#E1:162"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="txt" href="#E1:162_3"><span class="lab"><font color="Green" title="E71">A28</font></span></a>, <a class="txt" href="#E1:162_3_1"><span class="lab"><font color="Green" title="E72">A29</font></span></a>, <a class="ref" href="zf_lang1.html#D4" target="_self" title="ZF_LANG1:def.4">Def4</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><span class="kw">hence </span><a NAME="E3:162_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">z2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:162"><span class="lab"><font color="Green" title="E67">A22</font></span></a>, <a class="txt" href="#E19:162"><span class="lab"><font color="Green" title="E66">A21</font></span></a>, <a class="txt" href="#E9:162"><span class="lab"><font color="Green" title="E59">A6</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</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>
<div><span class="lab"><font color="Green" title="E71">A30</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="162_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:162_4"/><span class="lab"><font color="Green" title="E72">A31</font></span>: 
( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font> &amp; <font color="Maroon" title="c6">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font> )
 ; <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"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">z2</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="162_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">a</font> be    <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="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">z2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font> )</span><br/><span class="kw">assume </span><a NAME="E1:162_4_1"/><span class="lab"><font color="Green" title="E73">A32</font></span>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</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"><font color="Maroon" title="c5">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">z2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font></span><br/><a NAME="E2:162_4_1"/><span class="kw">then </span>
( <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:162"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:162_4_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">z2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:162"><span class="lab"><font color="Green" title="E65">A12</font></span></a>, <a class="txt" href="#E15:162"><span class="lab"><font color="Green" title="E63">A10</font></span></a>, <a class="txt" href="#E22:162"><span class="lab"><font color="Green" title="E69">A26</font></span></a>, <a class="txt" href="#E11:162"><span class="lab"><font color="Green" title="E60">A7</font></span></a>, <a class="txt" href="#E13:162"><span class="lab"><font color="Green" title="E61">A8</font></span></a>, <a class="txt" href="#E1:162"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="txt" href="#E1:162_4"><span class="lab"><font color="Green" title="E72">A31</font></span></a>, <a class="txt" href="#E1:162_4_1"><span class="lab"><font color="Green" title="E73">A32</font></span></a>, <a class="ref" href="zf_lang1.html#D4" target="_self" title="ZF_LANG1:def.4">Def4</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><span class="kw">hence </span><a NAME="E3:162_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">z2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:162"><span class="lab"><font color="Green" title="E67">A22</font></span></a>, <a class="txt" href="#E19:162"><span class="lab"><font color="Green" title="E66">A21</font></span></a>, <a class="txt" href="#E9:162"><span class="lab"><font color="Green" title="E59">A6</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="162_5"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:162_5"/><span class="lab"><font color="Green" title="E72">A33</font></span>: 
( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font> &amp; <font color="Maroon" title="c6">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x2</font> )
 ; <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"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">z2</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="162_5_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">a</font> be    <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="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">z2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font> )</span><br/><span class="kw">assume </span><a NAME="E1:162_5_1"/><span class="lab"><font color="Green" title="E73">A34</font></span>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</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"><font color="Maroon" title="c5">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">z2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font></span><br/><a NAME="E2:162_5_1"/><span class="kw">then </span>
( <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:162"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:162_5_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">z2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:162"><span class="lab"><font color="Green" title="E65">A12</font></span></a>, <a class="txt" href="#E15:162"><span class="lab"><font color="Green" title="E63">A10</font></span></a>, <a class="txt" href="#E22:162"><span class="lab"><font color="Green" title="E69">A26</font></span></a>, <a class="txt" href="#E11:162"><span class="lab"><font color="Green" title="E60">A7</font></span></a>, <a class="txt" href="#E13:162"><span class="lab"><font color="Green" title="E61">A8</font></span></a>, <a class="txt" href="#E1:162"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="txt" href="#E1:162_5"><span class="lab"><font color="Green" title="E72">A33</font></span></a>, <a class="txt" href="#E1:162_5_1"><span class="lab"><font color="Green" title="E73">A34</font></span></a>, <a class="ref" href="zf_lang1.html#D4" target="_self" title="ZF_LANG1:def.4">Def4</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><span class="kw">hence </span><a NAME="E3:162_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">z2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:162"><span class="lab"><font color="Green" title="E67">A22</font></span></a>, <a class="txt" href="#E19:162"><span class="lab"><font color="Green" title="E66">A21</font></span></a>, <a class="txt" href="#E9:162"><span class="lab"><font color="Green" title="E59">A6</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</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>
<span class="kw">hence </span><a NAME="E26:162"/>
( ( ( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x1</font> &amp; <font color="Maroon" title="c6">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x2</font> ) or ( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font> &amp; <font color="Maroon" title="c6">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x2</font> ) or ( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x1</font> &amp; <font color="Maroon" title="c6">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font> ) or ( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font> &amp; <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font> &amp; <font color="Maroon" title="c6">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font> ) ) implies <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">z2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:162"><span class="lab"><font color="Green" title="E68">A23</font></span></a>, <a class="txt" href="#E23:162"><span class="lab"><font color="Green" title="E70">A27</font></span></a>, <a class="txt" href="#E24:162"><span class="lab"><font color="Green" title="E71">A30</font></span></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>
