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<a NAME="E1:66_2_1"/><span class="lab"><font color="Green" title="E35">A5</font></span>: 
(  ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a> st <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Olive" title="b2">y</font> implies  ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ex <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a> st <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">k</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="zf_lang.html#K3" title="ZF_LANG:func.3">&lt;*</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="zf_lang.html#K3" title="ZF_LANG:func.3">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="zf_lang.html#K3" title="ZF_LANG:func.3">&lt;*</a><span class="default"><font color="Olive" title="b3">y</font></span><a href="zf_lang.html#K3" title="ZF_LANG:func.3">*&gt;</a></span> )
 
;<br/>
<a NAME="E2:66_2_1"/><span class="lab"><font color="Green" title="E36">A6</font></span>: 
(  ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a> st <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Olive" title="b2">y</font> implies  ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ex <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a> st <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">k</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="zf_lang.html#K3" title="ZF_LANG:func.3">&lt;*</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="zf_lang.html#K3" title="ZF_LANG:func.3">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="zf_lang.html#K3" title="ZF_LANG:func.3">&lt;*</a><span class="default"><font color="Olive" title="b3">y</font></span><a href="zf_lang.html#K3" title="ZF_LANG:func.3">*&gt;</a></span> )
 
;<br/>
<a NAME="E3:66_2_1"/>
( <font color="Maroon" title="c1">H</font> is  <a href="zf_lang.html#V2" title="ZF_LANG:attr.2">being_equality</a>  or <font color="Maroon" title="c1">H</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="txt" href="#E1:66"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="zf_lang.html#D15" target="_self" title="ZF_LANG:def.15">Def15</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a><span class="kw"> such that </span><br/><a NAME="E5:66_2_1"/><span class="lab"><font color="Green" title="E37">A7</font></span>: 
<font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">k</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="zf_lang.html#K3" title="ZF_LANG:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="zf_lang.html#K3" title="ZF_LANG:func.3">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="zf_lang.html#K3" title="ZF_LANG:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">y</font></span><a href="zf_lang.html#K3" title="ZF_LANG:func.3">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:66_2_1"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="txt" href="#E1:66_2_1"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="ref" href="zf_lang.html#D10" target="_self" title="ZF_LANG:def.10">Def10</a>, <a class="ref" href="zf_lang.html#D11" target="_self" title="ZF_LANG:def.11">Def11</a></span>;<br/>
<a NAME="E6:66_2_1"/>
<font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:66_2_1"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="ref" href="finseq_1.html#D10" title="FINSEQ_1:def.10">FINSEQ_1:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:66_2_1"/>
<font color="Maroon" title="c1">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 is   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</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/>


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