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<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</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> being  <a href="zf_refle.html#NM1" title="ZF_REFLE:NM.1">Subclass</a> of <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b2">E</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> is  <a href="zf_fund1.html#V8" title="ZF_FUND1:attr.8">closed_wrt_A1-A7</a>  &amp; <font color="Olive" title="b2">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> holds <br/>for <font color="Olive" title="b3">v1</font>, <font color="Olive" title="b4">v2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>  holds <br/> (  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Olive" title="b4">v2</font></span>)</span>,<font color="Olive" title="b2">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> &amp;  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Olive" title="b4">v2</font></span>)</span>,<font color="Olive" title="b2">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font> be   <a href="zf_refle.html#NM1" title="ZF_REFLE:NM.1">Subclass</a> of <font color="Maroon" title="c1">V</font>; <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">E</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon" title="c2">X</font> is  <a href="zf_fund1.html#V8" title="ZF_FUND1:attr.8">closed_wrt_A1-A7</a>  &amp; <font color="Olive" title="b1">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/>for <font color="Olive" title="b2">v1</font>, <font color="Olive" title="b3">v2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>  holds <br/> (  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Olive" title="b3">v2</font></span>)</span>,<font color="Olive" title="b1">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp;  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Olive" title="b3">v2</font></span>)</span>,<font color="Olive" title="b1">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">E</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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> is  <a href="zf_fund1.html#V8" title="ZF_FUND1:attr.8">closed_wrt_A1-A7</a>  &amp; <font color="Maroon" title="c3">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> implies for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>  holds <br/> (  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Olive" title="b2">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp;  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Olive" title="b2">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:43"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
<font color="Maroon" title="c2">X</font> is  <a href="zf_fund1.html#V8" title="ZF_FUND1:attr.8">closed_wrt_A1-A7</a> 
 <span class="kw">and </span><br/><a NAME="E2:43"/><span class="lab"><font color="Green" title="E47">A2</font></span>: 
<font color="Maroon" title="c3">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 ; <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">v1</font>, <font color="Olive" title="b2">v2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>  holds <br/> (  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Olive" title="b2">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp;  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Olive" title="b2">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )</span><br/>

<a NAME="E3:43"/><span class="lab"><font color="Green" title="E48">A3</font></span>: 
<font color="Maroon" title="c2">X</font> is  <a href="zf_fund1.html#V4" title="ZF_FUND1:attr.4">closed_wrt_A4</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="zf_fund1.html#D13" target="_self" title="ZF_FUND1:def.13">Def13</a></span>;<br/>
<a NAME="E4:43"/><span class="lab"><font color="Green" title="E49">A4</font></span>: 
<font color="Maroon" title="c2">X</font> is  <a href="zf_fund1.html#V1" title="ZF_FUND1:attr.1">closed_wrt_A1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="zf_fund1.html#D13" target="_self" title="ZF_FUND1:def.13">Def13</a></span>;<br/>
<a NAME="E5:43"/><span class="lab"><font color="Green" title="E50">A5</font></span>: 
 <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="zf_fund1.html#T7" target="_self" title="ZF_FUND1:th.7">Th7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">m</font> = <font color="Maroon" title="c3">E</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E47">A2</font></span></a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">v1</font>, <font color="Maroon" title="c6">v2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp;  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c7">H</font> = <font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">H9</font> = <font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font>;<br/>
<a NAME="E7:43"/><span class="lab"><font color="Green" title="E51">A6</font></span>: 
 <a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T63" title="ZF_LANG1:th.63">ZF_LANG1:63</a></span>;<br/>
<a NAME="E8:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A7</font></span>: 
<font color="Maroon" title="c5">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E9:43"/><span class="lab"><font color="Green" title="E53">A8</font></span>: 
<font color="Maroon" title="c6">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E10:43"/><span class="lab"><font color="Green" title="E54">A9</font></span>: 
 <a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T64" title="ZF_LANG1:th.64">ZF_LANG1:64</a></span>;<br/>
<a NAME="E11:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A10</font></span>: 
<font color="Maroon" title="c5">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E12:43"/><span class="lab"><font color="Green" title="E56">A11</font></span>: 
<font color="Maroon" title="c6">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E54">A9</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_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="E1:43_1"/>
( <font color="Maroon" title="c5">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v2</font> or <font color="Maroon" title="c5">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">v2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1"><span class="kw">suppose </span></a><a NAME="E1:43_1_1"/><span class="lab"><font color="Green" title="E57">A12</font></span>: 
<font color="Maroon" title="c5">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp;  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c9">a</font> =  <a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c10">Z</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> where <font color="Olive" title="b1">z</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">m</font> ) </span>}</span> ;<br/><a NAME="E2:43_1_1"/><span class="lab"><font color="Green" title="E58">A13</font></span>: 
 <a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="txt" href="#E1:43_1_1"><span class="lab"><font color="Green" title="E57">A12</font></span></a>, <a class="ref" href="enumset1.html#T69" title="ENUMSET1:th.69">ENUMSET1:69</a></span>;<br/><a NAME="E3:43_1_1"/><span class="lab"><font color="Green" title="E59">A14</font></span>: 
 <a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c5">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E4:43_1_1"/><span class="lab"><font color="Green" title="E60">A15</font></span>: 
<span class="p1">{<span class="default"> <span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> where <font color="Olive" title="b1">z</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">m</font> ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:43_1_1_1"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> where <font color="Olive" title="b1">z</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">m</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font>
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> where <font color="Olive" title="b1">z</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">m</font> ) </span>}</span> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> where <font color="Olive" title="b1">z</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">m</font> ) </span>}</span>  or <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font> )</span><br/>

<a NAME="E1:43_1_1_1_1"/><span class="lab"><font color="Green" title="E61">A16</font></span>: 
<font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font> is  <a href="zf_lang.html#V2" title="ZF_LANG:attr.2">being_equality</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/>
<span class="kw">assume </span><a NAME="E2:43_1_1_1_1"/>
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> where <font color="Olive" title="b1">z</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">m</font> ) </span>}</span> 
 ; <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="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">z</font>, <font color="Maroon" title="c13">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E4:43_1_1_1_1"/><span class="lab"><font color="Green" title="E62">A17</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p2"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><font color="Maroon" title="c12">z</font>,<font color="Maroon" title="c13">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E5:43_1_1_1_1"/><span class="lab"><font color="Green" title="E63">A18</font></span>: 
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:43_1_1_1_1"/><span class="lab"><font color="Green" title="E64">A19</font></span>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">m</font>
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">f</font> = <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>  <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c13">y</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<font color="Maroon" title="c3">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_1_1_1_1"><span class="lab"><font color="Green" title="E64">A19</font></span></a>, <a class="ref" href="funcop_1.html#T57" title="FUNCOP_1:th.57">FUNCOP_1:57</a></span>;<br/>
<a NAME="E8:43_1_1_1_1"/>
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c5">v1</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_1_1"><span class="lab"><font color="Green" title="E58">A13</font></span></a>, <a class="txt" href="#E5:43_1_1_1_1"><span class="lab"><font color="Green" title="E63">A18</font></span></a>, <a class="txt" href="zf_fund1.html#E19"><span class="lab"><font color="Green" title="E18">Lm6</font></span></a></span>;<br/>
<a NAME="E9:43_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A20</font></span>: 
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c5">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span></span>)</span></span>)</span> = 
 <a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="zf_fund1.html#E16"><span class="lab"><font color="Green" title="E15">Lm3</font></span></a></span>
<br/>.= 
<span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c12">z</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_1_1"><span class="lab"><font color="Green" title="E58">A13</font></span></a>, <a class="txt" href="#E9:43_1_1_1_1"><span class="lab"><font color="Green" title="E65">A20</font></span></a>, <a class="txt" href="zf_fund1.html#E19"><span class="lab"><font color="Green" title="E18">Lm6</font></span></a></span>
;<br/>
<a NAME="E11:43_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">z</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c14">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p0">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">z</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="grfunc_1.html#T18" title="GRFUNC_1:th.18">GRFUNC_1:18</a></span>;<br/>
<a NAME="E12:43_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">z</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">v1</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="txt" href="#E9:43_1_1_1_1"><span class="lab"><font color="Green" title="E65">A20</font></span></a>, <a class="txt" href="zf_fund1.html#E22"><span class="lab"><font color="Green" title="E21">Lm9</font></span></a></span>;<br/>
<a NAME="E13:43_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A21</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_1_1_1_1"><span class="lab"><font color="Green" title="E62">A17</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<font color="Maroon" title="c14">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c14">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1_1"><span class="lab"><font color="Green" title="E57">A12</font></span></a>, <a class="ref" href="zf_lang1.html#T1" title="ZF_LANG1:th.1">ZF_LANG1:1</a></span>
<br/>.= 
<font color="Maroon" title="c14">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T1" title="ZF_LANG1:th.1">ZF_LANG1:1</a></span>
;<br/>
<a NAME="E15:43_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1_1_1_1"><span class="lab"><font color="Green" title="E61">A16</font></span></a>, <a class="ref" href="zf_model.html#T7" title="ZF_MODEL:th.7">ZF_MODEL:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:43_1_1_1_1"/>
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43_1_1_1_1"><span class="lab"><font color="Green" title="E66">A21</font></span></a>, <a class="ref" href="zf_fund1.html#D5" target="_self" title="ZF_FUND1: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>

<span class="kw">reconsider </span><font color="Maroon" title="c11">z</font> =  <a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c5">v1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1_1"><span class="lab"><font color="Green" title="E59">A14</font></span></a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c12">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font> or <font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> where <font color="Olive" title="b1">z</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">m</font> ) </span>}</span>  )</span><br/>

<span class="kw">assume </span><a NAME="E3:43_1_1_1"/>
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</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="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> where <font color="Olive" title="b1">z</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">m</font> ) </span>}</span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<font color="Maroon" title="c3">E</font><span class="kw"> such that </span><br/><a NAME="E5:43_1_1_1"/><span class="lab"><font color="Green" title="E61">A22</font></span>: 
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:43_1_1_1"/>
<font color="Maroon" title="c13">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_fund1.html#D5" target="_self" title="ZF_FUND1:def.5">Def5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">y</font> = <font color="Maroon" title="c13">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">v1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="ref" href="zf_fund1.html#T1" target="_self" title="ZF_FUND1:th.1">Th1</a></span>;<br/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span></span>)</span></span>)</span> = 
 <a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="zf_fund1.html#E16"><span class="lab"><font color="Green" title="E15">Lm3</font></span></a></span>
<br/>.= 
<span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c11">z</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_1_1"><span class="lab"><font color="Green" title="E58">A13</font></span></a>, <a class="txt" href="zf_fund1.html#E19"><span class="lab"><font color="Green" title="E18">Lm6</font></span></a></span>
;<br/>
<a NAME="E9:43_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">z</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c13">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p0">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">z</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="grfunc_1.html#T18" title="GRFUNC_1:th.18">GRFUNC_1:18</a></span>;<br/>
<a NAME="E10:43_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A23</font></span>: 
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p2"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><font color="Maroon" title="c11">z</font>,<font color="Maroon" title="c14">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="txt" href="#E5:43_1_1_1"><span class="lab"><font color="Green" title="E61">A22</font></span></a>, <a class="txt" href="zf_fund1.html#E22"><span class="lab"><font color="Green" title="E21">Lm9</font></span></a></span>;<br/>
<a NAME="E11:43_1_1_1"/>
<font color="Maroon" title="c11">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c11">z</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E12:43_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_1_1"><span class="lab"><font color="Green" title="E58">A13</font></span></a>, <a class="txt" href="zf_fund1.html#E19"><span class="lab"><font color="Green" title="E18">Lm6</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:43_1_1_1"/>
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> where <font color="Olive" title="b1">z</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">m</font> ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43_1_1_1"><span class="lab"><font color="Green" title="E62">A23</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><span class="kw">end;</span></div><a NAME="E5:43_1_1"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c5">v1</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E3:43_1_1"><span class="lab"><font color="Green" title="E59">A14</font></span></a>, <a class="ref" href="zf_fund1.html#T2" target="_self" title="ZF_FUND1:th.2">Th2</a></span>;<br/><a NAME="E6:43_1_1"/><span class="kw">then </span>
 <a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_1_1"><span class="lab"><font color="Green" title="E58">A13</font></span></a>, <a class="txt" href="zf_fund1.html#E19"><span class="lab"><font color="Green" title="E18">Lm6</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:43_1_1"/>
 <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="txt" href="#E4:43_1_1"><span class="lab"><font color="Green" title="E60">A15</font></span></a>, <a class="ref" href="zf_fund1.html#D9" target="_self" title="ZF_FUND1:def.9">Def9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font></span><br/><a NAME="E8:43_1_1"/>
 <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c11">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font>;<br/>
<span class="kw">assume </span><a NAME="E2:43_1_1_2"/>
not  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<font color="Maroon" title="c3">E</font><span class="kw"> such that </span><br/><a NAME="E4:43_1_1_2"/>
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:43_1_1_2"/><span class="lab"><font color="Green" title="E61">A24</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_fund1.html#D5" target="_self" title="ZF_FUND1:def.5">Def5</a></span>;<br/>
<a NAME="E6:43_1_1_2"/>
<font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</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="E7:43_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43_1_1_2"><span class="lab"><font color="Green" title="E61">A24</font></span></a>, <a class="ref" href="zf_model.html#T8" title="ZF_MODEL:th.8">ZF_MODEL:8</a></span>;<br/>
<a NAME="E8:43_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T2" title="ZF_LANG1:th.2">ZF_LANG1:2</a></span>;<br/>
<a NAME="E9:43_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T2" title="ZF_LANG1:th.2">ZF_LANG1:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:43_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1_1"><span class="lab"><font color="Green" title="E57">A12</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><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E9:43_1_1"/>
 <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="zf_fund1.html#T3" target="_self" title="ZF_FUND1:th.3">Th3</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="43_1_2"><span class="kw">suppose </span></a><a NAME="E1:43_1_2"/><span class="lab"><font color="Green" title="E57">A25</font></span>: 
<font color="Maroon" title="c5">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">v2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp;  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )</span><br/><div class="add"><a NAME="E2:43_1_2"/>
(  <a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c6">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp;  <a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E3:43_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A26</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<span class="p2">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="zf_fund1.html#T6" target="_self" title="ZF_FUND1:th.6">Th6</a></span>;<br/><a NAME="E4:43_1_2"/>
(  <a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c5">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp;  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="txt" href="zf_fund1.html#E36"><span class="lab"><font color="Green" title="E35">Lm13</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E5:43_1_2"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c5">v1</font></span>)</span>,<span class="p2">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="zf_fund1.html#T6" target="_self" title="ZF_FUND1:th.6">Th6</a></span>;<br/><a NAME="E6:43_1_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c5">v1</font></span>)</span>,<span class="p3">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<span class="p3">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E3:43_1_2"><span class="lab"><font color="Green" title="E58">A26</font></span></a>, <a class="ref" href="zf_fund1.html#T6" target="_self" title="ZF_FUND1:th.6">Th6</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">d</font> = <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c5">v1</font></span>)</span>,<span class="p3">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<span class="p3">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> ;<br/><span class="kw">set </span><font color="Maroon" title="c10">fs</font> =  <a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span>;<br/><a NAME="E8:43_1_2"/><span class="lab"><font color="Green" title="E59">A27</font></span>: 
 <a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c5">v1</font></span>)</span>,<span class="p2">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c6">v2</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="txt" href="zf_fund1.html#E20"><span class="lab"><font color="Green" title="E19">Lm7</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E60">A28</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:43_1_2_1"/>
 <a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c5">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c6">v2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><font color="Maroon" title="c5">v1</font> = 
 <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c6">v2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_fund1.html#D3" target="_self" title="ZF_FUND1:def.3">Def3</a></span>
<br/>.= 
<font color="Maroon" title="c6">v2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_fund1.html#D3" target="_self" title="ZF_FUND1:def.3">Def3</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:43_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1_2"><span class="lab"><font color="Green" title="E57">A25</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><span class="kw">end;</span></div><a NAME="E10:43_1_2"/><span class="lab"><font color="Green" title="E61">A29</font></span>: 
<font color="Maroon" title="c9">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span></span>)</span>,<a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c11">g</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c5">v1</font></span>)</span>,<span class="p3">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>, <font color="Maroon" title="c12">h</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<span class="p3">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="grfunc_1.html#T15" title="GRFUNC_1:th.15">GRFUNC_1:15</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">e</font> = <font color="Maroon" title="c9">d</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43_1_2"><span class="lab"><font color="Green" title="E60">A28</font></span></a>, <a class="ref" href="grfunc_1.html#T19" title="GRFUNC_1:th.19">GRFUNC_1:19</a></span>;<br/>
<a NAME="E3:43_1_2_2"/><span class="lab"><font color="Green" title="E62">A30</font></span>: 
 <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E4:43_1_2_2"/><span class="lab"><font color="Green" title="E63">A31</font></span>: 
<font color="Maroon" title="c13">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T41" title="ENUMSET1:th.41">ENUMSET1:41</a></span>;<br/>
<span class="kw">then </span> <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">e</font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">g</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">h</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T26" title="RELAT_1:th.26">RELAT_1:26</a></span>
<br/>.= 
<span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">h</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T23" title="RELAT_1:th.23">RELAT_1:23</a></span>
<br/>.= 
<span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T23" title="RELAT_1:th.23">RELAT_1:23</a></span>
<br/>.= 
<span class="p1"><a href="classes2.html#K5" title="CLASSES2:func.5">{</a><span class="default"><span class="p2">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="classes2.html#K5" title="CLASSES2:func.5">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T41" title="ENUMSET1:th.41">ENUMSET1:41</a></span>
;<br/>
<a NAME="E6:43_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A32</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1_2_2"><span class="lab"><font color="Green" title="E62">A30</font></span></a>, <a class="ref" href="zfmisc_1.html#T38" title="ZFMISC_1:th.38">ZFMISC_1:38</a></span>;<br/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">e</font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">g</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">h</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_1_2_2"><span class="lab"><font color="Green" title="E63">A31</font></span></a>, <a class="ref" href="relat_1.html#T13" title="RELAT_1:th.13">RELAT_1:13</a></span>
<br/>.= 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c5">v1</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">h</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T23" title="RELAT_1:th.23">RELAT_1:23</a></span>
<br/>.= 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c5">v1</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c6">v2</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T23" title="RELAT_1:th.23">RELAT_1:23</a></span>
<br/>.= 
 <a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43_1_2"><span class="lab"><font color="Green" title="E59">A27</font></span></a>, <a class="ref" href="enumset1.html#T41" title="ENUMSET1:th.41">ENUMSET1:41</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:43_1_2_2"/>
<font color="Maroon" title="c9">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span></span>)</span>,<a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_1_2_2"><span class="lab"><font color="Green" title="E64">A32</font></span></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</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">set </span><font color="Maroon" title="c11">a</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="classes2.html#K5" title="CLASSES2:func.5">{</a><span class="default"><span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b1">x</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span>,<span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b1">x</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K5" title="CLASSES2:func.5">}</a></span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">m</font> </span>}</span> ;<br/><a NAME="E11:43_1_2"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="classes2.html#K5" title="CLASSES2:func.5">{</a><span class="default"><span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b1">x</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span>,<span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b1">x</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K5" title="CLASSES2:func.5">}</a></span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">m</font> </span>}</span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="ref" href="zf_fund1.html#T17" target="_self" title="ZF_FUND1:th.17">Th17</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">a</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="classes2.html#K5" title="CLASSES2:func.5">{</a><span class="default"><span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b1">x</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span>,<span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b1">x</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K5" title="CLASSES2:func.5">}</a></span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">m</font> </span>}</span>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> ;<br/><span class="kw">set </span><font color="Maroon" title="c13">b</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="classes2.html#K5" title="CLASSES2:func.5">{</a><span class="default"><span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b1">x</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span>,<span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b2">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K5" title="CLASSES2:func.5">}</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">m</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">m</font> ) </span>}</span> ;<br/><span class="kw">set </span><font color="Maroon" title="c14">Y</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c9">d</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">a</font> </span>}</span> ;<br/><a NAME="E13:43_1_2"/><span class="lab"><font color="Green" title="E62">A33</font></span>: 
<span class="p1">{<span class="default"> <span class="p2"><a href="classes2.html#K5" title="CLASSES2:func.5">{</a><span class="default"><span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b1">x</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span>,<span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b2">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K5" title="CLASSES2:func.5">}</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">m</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">m</font> ) </span>}</span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E4:43"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="ref" href="zf_fund1.html#D6" target="_self" title="ZF_FUND1:def.6">Def6</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">b</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="classes2.html#K5" title="CLASSES2:func.5">{</a><span class="default"><span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b1">x</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span>,<span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b2">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K5" title="CLASSES2:func.5">}</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">m</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">m</font> ) </span>}</span>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> ;<br/><span class="kw">set </span><font color="Maroon" title="c16">Z</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c9">d</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">b</font> </span>}</span> ;<br/><a NAME="E15:43_1_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c9">d</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">a</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:43_1_2_3"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c9">d</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">a</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font>
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c9">d</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">a</font> </span>}</span> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c17">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c9">d</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">a</font> </span>}</span>  or <font color="Maroon" title="c17">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:43_1_2_3_1"/>
<font color="Maroon" title="c17">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c9">d</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">a</font> </span>}</span> 
 ; <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="c17">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E3:43_1_2_3_1"/><span class="lab"><font color="Green" title="E63">A34</font></span>: 
<font color="Maroon" title="c17">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">d</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Maroon" title="c18">x</font>
 <span class="kw">and </span><br/><a NAME="E4:43_1_2_3_1"/><span class="lab"><font color="Green" title="E64">A35</font></span>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">a</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E6:43_1_2_3_1"/><span class="lab"><font color="Green" title="E65">A36</font></span>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="classes2.html#K5" title="CLASSES2:func.5">{</a><span class="default"><span class="p2"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c19">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span>,<span class="p2"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c19">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K5" title="CLASSES2:func.5">}</a></span>
 <span class="kw">and </span><br/><a NAME="E7:43_1_2_3_1"/><span class="lab"><font color="Green" title="E66">A37</font></span>: 
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_1_2_3_1"><span class="lab"><font color="Green" title="E64">A35</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">f</font> = <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>  <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c19">y</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<font color="Maroon" title="c3">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43_1_2_3_1"><span class="lab"><font color="Green" title="E66">A37</font></span></a>, <a class="ref" href="funcop_1.html#T57" title="FUNCOP_1:th.57">FUNCOP_1:57</a></span>;<br/>
<span class="lab"><font color="Green" title="E67">A38</font></span>: <font color="Maroon" title="c20">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c19">y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c20">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
;<br/>
<a NAME="E10:43_1_2_3_1"/>
<font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font> is  <a href="zf_lang.html#V2" title="ZF_LANG:attr.2">being_equality</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="E11:43_1_2_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A39</font></span>: 
<font color="Maroon" title="c20">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43_1_2_3_1"><span class="lab"><font color="Green" title="E67">A38</font></span></a>, <a class="ref" href="zf_model.html#T7" title="ZF_MODEL:th.7">ZF_MODEL:7</a></span>;<br/>
<span class="lab"><font color="Green" title="E69">A40</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c6">v2</font></span>)</span> = 
<font color="Maroon" title="c20">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43"><span class="lab"><font color="Green" title="E53">A8</font></span></a>, <a class="txt" href="zf_fund1.html#E22"><span class="lab"><font color="Green" title="E21">Lm9</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c19">y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
;<br/>
<span class="lab"><font color="Green" title="E70">A41</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c5">v1</font></span>)</span> = 
<font color="Maroon" title="c20">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">v1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="txt" href="zf_fund1.html#E22"><span class="lab"><font color="Green" title="E21">Lm9</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c19">y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
;<br/>
<a NAME="E14:43_1_2_3_1"/><span class="lab"><font color="Green" title="E71">A42</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c5">v1</font></span>)</span>,<span class="p2">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c6">v2</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43_1_2"><span class="lab"><font color="Green" title="E59">A27</font></span></a>, <a class="txt" href="zf_fund1.html#E16"><span class="lab"><font color="Green" title="E15">Lm3</font></span></a></span>;<br/>
<a NAME="E15:43_1_2_3_1"/>
<font color="Maroon" title="c17">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c5">v1</font></span>)</span>,<font color="Maroon" title="c19">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c19">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1_2_3_1"><span class="lab"><font color="Green" title="E63">A34</font></span></a>, <a class="txt" href="#E6:43_1_2_3_1"><span class="lab"><font color="Green" title="E65">A36</font></span></a>, <a class="txt" href="zf_fund1.html#E45"><span class="lab"><font color="Green" title="E43">Lm15</font></span></a></span>;<br/>
<a NAME="E16:43_1_2_3_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:43_1_2_3_1"><span class="lab"><font color="Green" title="E71">A42</font></span></a>, <a class="txt" href="#E13:43_1_2_3_1"><span class="lab"><font color="Green" title="E70">A41</font></span></a>, <a class="txt" href="#E12:43_1_2_3_1"><span class="lab"><font color="Green" title="E69">A40</font></span></a>, <a class="txt" href="zf_fund1.html#E46"><span class="lab"><font color="Green" title="E44">Lm16</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:43_1_2_3_1"/>
<font color="Maroon" title="c17">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:43_1_2_3_1"><span class="lab"><font color="Green" title="E68">A39</font></span></a>, <a class="ref" href="zf_fund1.html#D5" target="_self" title="ZF_FUND1: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>

<span class="kw">let </span><font color="Maroon" title="c17">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c17">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font> or <font color="Maroon" title="c17">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c9">d</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">a</font> </span>}</span>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:43_1_2_3"/>
<font color="Maroon" title="c17">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</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="c17">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c9">d</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">a</font> </span>}</span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<font color="Maroon" title="c3">E</font><span class="kw"> such that </span><br/><a NAME="E4:43_1_2_3"/><span class="lab"><font color="Green" title="E63">A43</font></span>: 
<font color="Maroon" title="c17">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:43_1_2_3"/><span class="lab"><font color="Green" title="E64">A44</font></span>: 
<font color="Maroon" title="c18">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_fund1.html#D5" target="_self" title="ZF_FUND1:def.5">Def5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">y</font> = <font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">v1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="ref" href="zf_fund1.html#T1" target="_self" title="ZF_FUND1:th.1">Th1</a></span>;<br/>
<a NAME="E7:43_1_2_3"/>
<font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font> is  <a href="zf_lang.html#V2" title="ZF_LANG:attr.2">being_equality</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="E8:43_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43_1_2_3"><span class="lab"><font color="Green" title="E64">A44</font></span></a>, <a class="ref" href="zf_model.html#T7" title="ZF_MODEL:th.7">ZF_MODEL:7</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">v1</font> = 
<font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T1" title="ZF_LANG1:th.1">ZF_LANG1:1</a></span>
<br/>.= 
<font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T1" title="ZF_LANG1:th.1">ZF_LANG1:1</a></span>
;<br/>
<a NAME="E10:43_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A45</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c6">v2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43"><span class="lab"><font color="Green" title="E53">A8</font></span></a>, <a class="txt" href="zf_fund1.html#E22"><span class="lab"><font color="Green" title="E21">Lm9</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">x</font> = <span class="p1"><a href="classes2.html#K5" title="CLASSES2:func.5">{</a><span class="default"><span class="p2"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c19">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span>,<span class="p2"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c19">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K5" title="CLASSES2:func.5">}</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<a NAME="E12:43_1_2_3"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c5">v1</font></span>)</span>,<span class="p2">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c6">v2</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c5">v1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="txt" href="#E8:43_1_2"><span class="lab"><font color="Green" title="E59">A27</font></span></a>, <a class="txt" href="zf_fund1.html#E16"><span class="lab"><font color="Green" title="E15">Lm3</font></span></a>, <a class="txt" href="zf_fund1.html#E22"><span class="lab"><font color="Green" title="E21">Lm9</font></span></a></span>;<br/>
<a NAME="E13:43_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c17">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c5">v1</font></span>)</span>,<font color="Maroon" title="c19">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c19">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_1_2_3"><span class="lab"><font color="Green" title="E63">A43</font></span></a>, <a class="txt" href="#E10:43_1_2_3"><span class="lab"><font color="Green" title="E65">A45</font></span></a>, <a class="txt" href="zf_fund1.html#E46"><span class="lab"><font color="Green" title="E44">Lm16</font></span></a></span>;<br/>
<a NAME="E14:43_1_2_3"/><span class="kw">then </span>
( <span class="p1"><a href="classes2.html#K5" title="CLASSES2:func.5">{</a><span class="default"><span class="p2"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c19">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span>,<span class="p2"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c19">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K5" title="CLASSES2:func.5">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">a</font> &amp; <font color="Maroon" title="c17">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">d</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Maroon" title="c20">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="zf_fund1.html#E45"><span class="lab"><font color="Green" title="E43">Lm15</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:43_1_2_3"/>
<font color="Maroon" title="c17">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c9">d</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">a</font> </span>}</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="E16:43_1_2"/>
 <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E10:43_1_2"><span class="lab"><font color="Green" title="E61">A29</font></span></a>, <a class="ref" href="zf_fund1.html#T10" target="_self" title="ZF_FUND1:th.10">Th10</a>, <a class="ref" href="zf_fund1.html#T17" target="_self" title="ZF_FUND1:th.17">Th17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font></span><br/><a NAME="E17:43_1_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c9">d</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">b</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:43_1_2_4"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c9">d</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">b</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font>
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c9">d</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">b</font> </span>}</span> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c17">a1</font> = <font color="Maroon" title="c5">v1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>  ;<br/>
<span class="kw">let </span><font color="Maroon" title="c18">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c18">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c9">d</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">b</font> </span>}</span>  or <font color="Maroon" title="c18">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:43_1_2_4_1"/>
<font color="Maroon" title="c18">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c9">d</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">b</font> </span>}</span> 
 ; <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="c18">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E4:43_1_2_4_1"/><span class="lab"><font color="Green" title="E63">A46</font></span>: 
<font color="Maroon" title="c18">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">d</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Maroon" title="c19">x</font>
 <span class="kw">and </span><br/><a NAME="E5:43_1_2_4_1"/><span class="lab"><font color="Green" title="E64">A47</font></span>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">b</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c20">y</font>, <font color="Maroon" title="c21">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E7:43_1_2_4_1"/><span class="lab"><font color="Green" title="E65">A48</font></span>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="classes2.html#K5" title="CLASSES2:func.5">{</a><span class="default"><span class="p2"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c20">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span>,<span class="p2"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c21">z</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K5" title="CLASSES2:func.5">}</a></span>
 <span class="kw">and </span><br/><a NAME="E8:43_1_2_4_1"/><span class="lab"><font color="Green" title="E66">A49</font></span>: 
<font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">z</font>
 <span class="kw">and </span><br/><a NAME="E9:43_1_2_4_1"/><span class="lab"><font color="Green" title="E67">A50</font></span>: 
( <font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">m</font> &amp; <font color="Maroon" title="c21">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">m</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43_1_2_4_1"><span class="lab"><font color="Green" title="E64">A47</font></span></a></span>;<br/>
<a NAME="E10:43_1_2_4_1"/><span class="lab"><font color="Green" title="E68">A51</font></span>: 
<font color="Maroon" title="c18">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c5">v1</font></span>)</span>,<font color="Maroon" title="c20">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c21">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_1_2_4_1"><span class="lab"><font color="Green" title="E63">A46</font></span></a>, <a class="txt" href="#E7:43_1_2_4_1"><span class="lab"><font color="Green" title="E65">A48</font></span></a>, <a class="txt" href="zf_fund1.html#E45"><span class="lab"><font color="Green" title="E43">Lm15</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c22">y9</font> = <font color="Maroon" title="c20">y</font>, <font color="Maroon" title="c23">z9</font> = <font color="Maroon" title="c21">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43_1_2_4_1"><span class="lab"><font color="Green" title="E67">A50</font></span></a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">E</font> = <font color="Maroon" title="c23">z9</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c24">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<font color="Maroon" title="c3">E</font><span class="kw"> such that </span><br/><a NAME="E13:43_1_2_4_1"/><span class="lab"><font color="Green" title="E69">A52</font></span>: 
( <font color="Maroon" title="c24">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">y9</font> &amp; ( for <font color="Olive" title="b1">v3</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>   st <font color="Olive" title="b1">v3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">a1</font> holds <br/><font color="Maroon" title="c24">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">v3</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S6" title="FUNCT_2:sch.6">FUNCT_2:sch 6</a>();<br/></span>
<span class="lab"><font color="Green" title="E70">A53</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c6">v2</font></span>)</span> = 
<font color="Maroon" title="c24">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43"><span class="lab"><font color="Green" title="E56">A11</font></span></a>, <a class="txt" href="zf_fund1.html#E22"><span class="lab"><font color="Green" title="E21">Lm9</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c21">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1_2"><span class="lab"><font color="Green" title="E57">A25</font></span></a>, <a class="txt" href="#E13:43_1_2_4_1"><span class="lab"><font color="Green" title="E69">A52</font></span></a></span>
;<br/>
<a NAME="E15:43_1_2_4_1"/><span class="lab"><font color="Green" title="E71">A54</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c5">v1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:43"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="txt" href="#E13:43_1_2_4_1"><span class="lab"><font color="Green" title="E69">A52</font></span></a>, <a class="txt" href="zf_fund1.html#E22"><span class="lab"><font color="Green" title="E21">Lm9</font></span></a></span>;<br/>
<a NAME="E16:43_1_2_4_1"/>
<font color="Maroon" title="c24">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1_2"><span class="lab"><font color="Green" title="E57">A25</font></span></a>, <a class="txt" href="#E8:43_1_2_4_1"><span class="lab"><font color="Green" title="E66">A49</font></span></a>, <a class="txt" href="#E13:43_1_2_4_1"><span class="lab"><font color="Green" title="E69">A52</font></span></a></span>;<br/>
<a NAME="E17:43_1_2_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c24">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T2" title="ZF_LANG1:th.2">ZF_LANG1:2</a></span>;<br/>
<a NAME="E18:43_1_2_4_1"/><span class="kw">then </span>
( <font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font> is  <a href="zf_lang.html#V3" title="ZF_LANG:attr.3">being_membership</a>  &amp; <font color="Maroon" title="c24">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span> )
 
<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>, <a class="ref" href="zf_lang1.html#T2" title="ZF_LANG1:th.2">ZF_LANG1:2</a></span>;<br/>
<a NAME="E19:43_1_2_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A55</font></span>: 
<font color="Maroon" title="c24">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_model.html#T8" title="ZF_MODEL:th.8">ZF_MODEL:8</a></span>;<br/>
<a NAME="E20:43_1_2_4_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c5">v1</font></span>)</span>,<span class="p2">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c6">v2</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E54">A9</font></span></a>, <a class="txt" href="#E8:43_1_2"><span class="lab"><font color="Green" title="E59">A27</font></span></a>, <a class="txt" href="zf_fund1.html#E16"><span class="lab"><font color="Green" title="E15">Lm3</font></span></a></span>;<br/>
<a NAME="E21:43_1_2_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43_1_2_4_1"><span class="lab"><font color="Green" title="E68">A51</font></span></a>, <a class="txt" href="#E15:43_1_2_4_1"><span class="lab"><font color="Green" title="E71">A54</font></span></a>, <a class="txt" href="#E14:43_1_2_4_1"><span class="lab"><font color="Green" title="E70">A53</font></span></a>, <a class="txt" href="zf_fund1.html#E46"><span class="lab"><font color="Green" title="E44">Lm16</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E22:43_1_2_4_1"/>
<font color="Maroon" title="c18">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:43_1_2_4_1"><span class="lab"><font color="Green" title="E72">A55</font></span></a>, <a class="ref" href="zf_fund1.html#D5" target="_self" title="ZF_FUND1: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>

<span class="kw">let </span><font color="Maroon" title="c17">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c17">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font> or <font color="Maroon" title="c17">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c9">d</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">b</font> </span>}</span>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:43_1_2_4"/>
<font color="Maroon" title="c17">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</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="c17">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c9">d</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">b</font> </span>}</span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<font color="Maroon" title="c3">E</font><span class="kw"> such that </span><br/><a NAME="E4:43_1_2_4"/><span class="lab"><font color="Green" title="E63">A56</font></span>: 
<font color="Maroon" title="c17">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:43_1_2_4"/><span class="lab"><font color="Green" title="E64">A57</font></span>: 
<font color="Maroon" title="c18">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_fund1.html#D5" target="_self" title="ZF_FUND1:def.5">Def5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">z</font> = <font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="ref" href="zf_fund1.html#T1" target="_self" title="ZF_FUND1:th.1">Th1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">y</font> = <font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">v1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="ref" href="zf_fund1.html#T1" target="_self" title="ZF_FUND1:th.1">Th1</a></span>;<br/>
<a NAME="E8:43_1_2_4"/>
<font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</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="E9:43_1_2_4"/><span class="kw">then </span>
<font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43_1_2_4"><span class="lab"><font color="Green" title="E64">A57</font></span></a>, <a class="ref" href="zf_model.html#T8" title="ZF_MODEL:th.8">ZF_MODEL:8</a></span>;<br/>
<a NAME="E10:43_1_2_4"/><span class="kw">then </span>
<font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T2" title="ZF_LANG1:th.2">ZF_LANG1:2</a></span>;<br/>
<a NAME="E11:43_1_2_4"/><span class="kw">then </span>
<font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T2" title="ZF_LANG1:th.2">ZF_LANG1:2</a></span>;<br/>
<a NAME="E12:43_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A58</font></span>: 
<span class="p1"><a href="classes2.html#K5" title="CLASSES2:func.5">{</a><span class="default"><span class="p2"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c20">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span>,<span class="p2"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c19">z</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K5" title="CLASSES2:func.5">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">b</font>
 
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">x</font> = <span class="p1"><a href="classes2.html#K5" title="CLASSES2:func.5">{</a><span class="default"><span class="p2"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c20">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span>,<span class="p2"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c19">z</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K5" title="CLASSES2:func.5">}</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<a NAME="E14:43_1_2_4"/><span class="lab"><font color="Green" title="E66">A59</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c6">v2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43"><span class="lab"><font color="Green" title="E56">A11</font></span></a>, <a class="txt" href="zf_fund1.html#E22"><span class="lab"><font color="Green" title="E21">Lm9</font></span></a></span>;<br/>
<a NAME="E15:43_1_2_4"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c5">v1</font></span>)</span>,<span class="p2">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c6">v2</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c5">v1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">v1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E54">A9</font></span></a>, <a class="txt" href="#E11:43"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="txt" href="#E8:43_1_2"><span class="lab"><font color="Green" title="E59">A27</font></span></a>, <a class="txt" href="zf_fund1.html#E16"><span class="lab"><font color="Green" title="E15">Lm3</font></span></a>, <a class="txt" href="zf_fund1.html#E22"><span class="lab"><font color="Green" title="E21">Lm9</font></span></a></span>;<br/>
<a NAME="E16:43_1_2_4"/><span class="kw">then </span>
<font color="Maroon" title="c17">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c5">v1</font></span>)</span>,<font color="Maroon" title="c20">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c19">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_1_2_4"><span class="lab"><font color="Green" title="E63">A56</font></span></a>, <a class="txt" href="#E14:43_1_2_4"><span class="lab"><font color="Green" title="E66">A59</font></span></a>, <a class="txt" href="zf_fund1.html#E46"><span class="lab"><font color="Green" title="E44">Lm16</font></span></a></span>;<br/>
<a NAME="E17:43_1_2_4"/><span class="kw">then </span>
<font color="Maroon" title="c17">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">d</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Maroon" title="c21">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="zf_fund1.html#E45"><span class="lab"><font color="Green" title="E43">Lm15</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:43_1_2_4"/>
<font color="Maroon" title="c17">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c9">d</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">b</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43_1_2_4"><span class="lab"><font color="Green" title="E65">A58</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><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E18:43_1_2"/>
 <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c6">v2</font></span>)</span>,<font color="Maroon" title="c3">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E13:43_1_2"><span class="lab"><font color="Green" title="E62">A33</font></span></a>, <a class="txt" href="#E10:43_1_2"><span class="lab"><font color="Green" title="E61">A29</font></span></a>, <a class="ref" href="zf_fund1.html#T10" target="_self" title="ZF_FUND1:th.10">Th10</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>

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