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<span class="kw">let </span><font color="Maroon" title="c1">E</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">E</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a>  <a href="zf_model.html#K6" title="ZF_MODEL:func.6">the_axiom_of_extensionality</a>  implies  ex <font color="Olive" title="b1">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b1">X</font> is  <a href="ordinal1.html#V1" title="ORDINAL1:attr.1">epsilon-transitive</a>  &amp; <font color="Maroon" title="c1">E</font>,<font color="Olive" title="b1">X</font> <a href="zf_colla.html#R2" title="ZF_COLLA:pred.2">are_epsilon-isomorphic</a>  ) )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c2">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E2:15"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font>
 <span class="kw">and </span><br/><a NAME="E3:15"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
 for <font color="Olive" title="b1">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> holds  <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Olive" title="b1">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_colla.html#T9" target="_self" onmouseover="rs('zf_colla/T9')" onmouseout="rh()">Th9</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:15"/><span class="lab"><font color="Green" title="E13">A3</font></span>: 
<font color="Maroon" title="c1">E</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a>  <a href="zf_model.html#K6" title="ZF_MODEL:func.6">the_axiom_of_extensionality</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b1">X</font> is  <a href="ordinal1.html#V1" title="ORDINAL1:attr.1">epsilon-transitive</a>  &amp; <font color="Maroon" title="c1">E</font>,<font color="Olive" title="b1">X</font> <a href="zf_colla.html#R2" title="ZF_COLLA:pred.2">are_epsilon-isomorphic</a>  )</span><br/>

<div><span class="lab"><font color="Green" title="E14">A4</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="15_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds <br/>(  not <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">f</font> or  not <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">f</font> or  not <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">b</font> or  not <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> )</span></a><div class="add"><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>] <span class="kw">means </span> ex <font color="Olive" title="b1">d1</font>, <font color="Olive" title="b2">d2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <br/>( <font color="Olive" title="b1">d1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,$1) &amp; <font color="Olive" title="b2">d2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,$1) &amp; <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">d2</font> &amp; <font color="Olive" title="b1">d1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">d2</font> );<br/><span class="kw">given </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><a NAME="E2:15_1"/><span class="lab"><font color="Green" title="E15">A5</font></span>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">f</font> )
 <span class="kw">and </span><br/><a NAME="E3:15_1"/><span class="lab"><font color="Green" title="E16">A6</font></span>: 
( <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c5">d1</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c6">d2</font> = <font color="Maroon" title="c4">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E2:15_1"><span class="lab"><font color="Green" title="E15">A5</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c7">A1</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E6:15_1"/><span class="lab"><font color="Green" title="E17">A7</font></span>: 
<font color="Maroon" title="c5">d1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c7">A1</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_colla.html#T5" target="_self" onmouseover="rs('zf_colla/T5')" onmouseout="rh()">Th5</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c8">A2</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E8:15_1"/><span class="lab"><font color="Green" title="E18">A8</font></span>: 
<font color="Maroon" title="c6">d2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c8">A2</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_colla.html#T5" target="_self" onmouseover="rs('zf_colla/T5')" onmouseout="rh()">Th5</a></span>;<br/><a NAME="E9:15_1"/>
( <font color="Maroon" title="c7">A1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">A2</font> or <font color="Maroon" title="c8">A2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c7">A1</font> )
 ;<br/><a NAME="E10:15_1"/><span class="kw">then </span>
(  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c7">A1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c8">A2</font>) or  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c8">A2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c7">A1</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_colla.html#T4" target="_self" onmouseover="rs('zf_colla/T4')" onmouseout="rh()">Th4</a></span>;<br/><a NAME="E11:15_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A9</font></span>: 
 ex <font color="Olive" title="b1">A</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="txt" href="#E6:15_1"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="txt" href="#E8:15_1"><span class="lab"><font color="Green" title="E18">A8</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">A</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E13:15_1"/><span class="lab"><font color="Green" title="E20">A10</font></span>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">A</font>] &amp; (  for <font color="Olive" title="b1">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B</font>] holds <br/><font color="Maroon" title="c9">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">B</font> ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="ordinal1.html#S1" onmouseover="rs('ordinal1/S1')" onmouseout="rh()">ORDINAL1:sch 1</a>(<a class="txt" href="#E11:15_1"><span class="lab"><font color="Green" title="E19">A9</font></span></a>);<br/></span><span class="kw">consider </span><font color="Maroon" title="c10">d1</font>, <font color="Maroon" title="c11">d2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font><span class="kw"> such that </span><br/><a NAME="E15:15_1"/><span class="lab"><font color="Green" title="E21">A11</font></span>: 
<font color="Maroon" title="c10">d1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c9">A</font>)
 <span class="kw">and </span><br/><a NAME="E16:15_1"/><span class="lab"><font color="Green" title="E22">A12</font></span>: 
<font color="Maroon" title="c11">d2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c9">A</font>)
 <span class="kw">and </span><br/><a NAME="E17:15_1"/><span class="lab"><font color="Green" title="E23">A13</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">d2</font>
 <span class="kw">and </span><br/><a NAME="E18:15_1"/><span class="lab"><font color="Green" title="E24">A14</font></span>: 
<font color="Maroon" title="c10">d1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">d2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:15_1"><span class="lab"><font color="Green" title="E20">A10</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font><span class="kw"> such that </span><br/><a NAME="E20:15_1"/><span class="lab"><font color="Green" title="E25">A15</font></span>: 
( ( <font color="Maroon" title="c12">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">d1</font> &amp;  not <font color="Maroon" title="c12">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">d2</font> ) or ( <font color="Maroon" title="c12">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">d2</font> &amp;  not <font color="Maroon" title="c12">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">d1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:15"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="txt" href="#E18:15_1"><span class="lab"><font color="Green" title="E24">A14</font></span></a>, <a class="ref" href="zf_colla.html#T11" target="_self" onmouseover="rs('zf_colla/T11')" onmouseout="rh()">Th11</a></span>;<br/><a NAME="E21:15_1"/><span class="lab"><font color="Green" title="E26">A16</font></span>: 
( <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c10">d1</font> &amp; <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c11">d2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E12">A2</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E27">A17</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="15_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">d2</font> implies <font color="Maroon" title="c12">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">d1</font> )</span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:15_1_1"/><span class="lab"><font color="Green" title="E28">A18</font></span>: 
<font color="Maroon" title="c12">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">d2</font>
 <span class="kw">and </span><br/><a NAME="E2:15_1_1"/><span class="lab"><font color="Green" title="E29">A19</font></span>: 
 not <font color="Maroon" title="c12">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">d1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">consider </span><font color="Maroon" title="c13">A1</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E4:15_1_1"/><span class="lab"><font color="Green" title="E30">A20</font></span>: 
( <font color="Maroon" title="c13">A1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c12">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c13">A1</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:15_1"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E1:15_1_1"><span class="lab"><font color="Green" title="E28">A18</font></span></a>, <a class="ref" href="zf_colla.html#T6" target="_self" onmouseover="rs('zf_colla/T6')" onmouseout="rh()">Th6</a></span>;<br/><a NAME="E5:15_1_1"/>
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c11">d2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E1:15_1_1"><span class="lab"><font color="Green" title="E28">A18</font></span></a>, <a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E7:15_1_1"/><span class="lab"><font color="Green" title="E31">A21</font></span>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">and </span><br/><a NAME="E8:15_1_1"/><span class="lab"><font color="Green" title="E32">A22</font></span>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">d1</font>
 <span class="kw">and </span><br/><a NAME="E9:15_1_1"/><span class="lab"><font color="Green" title="E33">A23</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:15_1"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="txt" href="#E21:15_1"><span class="lab"><font color="Green" title="E26">A16</font></span></a>, <a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">v</font> = <font color="Maroon" title="c14">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E7:15_1_1"><span class="lab"><font color="Green" title="E31">A21</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c16">A2</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E12:15_1_1"/><span class="lab"><font color="Green" title="E34">A24</font></span>: 
( <font color="Maroon" title="c16">A2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c15">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c16">A2</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:15_1"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="txt" href="#E8:15_1_1"><span class="lab"><font color="Green" title="E32">A22</font></span></a>, <a class="ref" href="zf_colla.html#T6" target="_self" onmouseover="rs('zf_colla/T6')" onmouseout="rh()">Th6</a></span>;<br/><a NAME="E13:15_1_1"/>
( <font color="Maroon" title="c13">A1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c16">A2</font> or <font color="Maroon" title="c16">A2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c13">A1</font> )
 ;<br/><a NAME="E14:15_1_1"/><span class="kw">then </span>
(  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c13">A1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c16">A2</font>) or  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c16">A2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c13">A1</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_colla.html#T4" target="_self" onmouseover="rs('zf_colla/T4')" onmouseout="rh()">Th4</a></span>;<br/><span class="kw">hence </span><a NAME="E15:15_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:15_1"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="txt" href="#E2:15_1_1"><span class="lab"><font color="Green" title="E29">A19</font></span></a>, <a class="txt" href="#E8:15_1_1"><span class="lab"><font color="Green" title="E32">A22</font></span></a>, <a class="txt" href="#E9:15_1_1"><span class="lab"><font color="Green" title="E33">A23</font></span></a>, <a class="txt" href="#E4:15_1_1"><span class="lab"><font color="Green" title="E30">A20</font></span></a>, <a class="txt" href="#E12:15_1_1"><span class="lab"><font color="Green" title="E34">A24</font></span></a>, <a class="ref" href="ordinal1.html#T5" onmouseover="rs('ordinal1/T5')" onmouseout="rh()">ORDINAL1:5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="15_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">d1</font> implies <font color="Maroon" title="c12">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">d2</font> )</span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:15_1_2"/><span class="lab"><font color="Green" title="E28">A25</font></span>: 
<font color="Maroon" title="c12">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">d1</font>
 <span class="kw">and </span><br/><a NAME="E2:15_1_2"/><span class="lab"><font color="Green" title="E29">A26</font></span>: 
 not <font color="Maroon" title="c12">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">d2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">consider </span><font color="Maroon" title="c13">A1</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E4:15_1_2"/><span class="lab"><font color="Green" title="E30">A27</font></span>: 
( <font color="Maroon" title="c13">A1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c12">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c13">A1</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:15_1"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="txt" href="#E1:15_1_2"><span class="lab"><font color="Green" title="E28">A25</font></span></a>, <a class="ref" href="zf_colla.html#T6" target="_self" onmouseover="rs('zf_colla/T6')" onmouseout="rh()">Th6</a></span>;<br/><a NAME="E5:15_1_2"/>
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c10">d1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E1:15_1_2"><span class="lab"><font color="Green" title="E28">A25</font></span></a>, <a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E7:15_1_2"/><span class="lab"><font color="Green" title="E31">A28</font></span>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">and </span><br/><a NAME="E8:15_1_2"/><span class="lab"><font color="Green" title="E32">A29</font></span>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">d2</font>
 <span class="kw">and </span><br/><a NAME="E9:15_1_2"/><span class="lab"><font color="Green" title="E33">A30</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:15_1"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="txt" href="#E21:15_1"><span class="lab"><font color="Green" title="E26">A16</font></span></a>, <a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">v</font> = <font color="Maroon" title="c14">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E7:15_1_2"><span class="lab"><font color="Green" title="E31">A28</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c16">A2</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E12:15_1_2"/><span class="lab"><font color="Green" title="E34">A31</font></span>: 
( <font color="Maroon" title="c16">A2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c15">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c16">A2</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:15_1"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E8:15_1_2"><span class="lab"><font color="Green" title="E32">A29</font></span></a>, <a class="ref" href="zf_colla.html#T6" target="_self" onmouseover="rs('zf_colla/T6')" onmouseout="rh()">Th6</a></span>;<br/><a NAME="E13:15_1_2"/>
( <font color="Maroon" title="c13">A1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c16">A2</font> or <font color="Maroon" title="c16">A2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c13">A1</font> )
 ;<br/><a NAME="E14:15_1_2"/><span class="kw">then </span>
(  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c13">A1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c16">A2</font>) or  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c16">A2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c13">A1</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_colla.html#T4" target="_self" onmouseover="rs('zf_colla/T4')" onmouseout="rh()">Th4</a></span>;<br/><span class="kw">hence </span><a NAME="E15:15_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:15_1"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="txt" href="#E2:15_1_2"><span class="lab"><font color="Green" title="E29">A26</font></span></a>, <a class="txt" href="#E8:15_1_2"><span class="lab"><font color="Green" title="E32">A29</font></span></a>, <a class="txt" href="#E9:15_1_2"><span class="lab"><font color="Green" title="E33">A30</font></span></a>, <a class="txt" href="#E4:15_1_2"><span class="lab"><font color="Green" title="E30">A27</font></span></a>, <a class="txt" href="#E12:15_1_2"><span class="lab"><font color="Green" title="E34">A31</font></span></a>, <a class="ref" href="ordinal1.html#T5" onmouseover="rs('ordinal1/T5')" onmouseout="rh()">ORDINAL1:5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E24:15_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:15_1"><span class="lab"><font color="Green" title="E25">A15</font></span></a>, <a class="txt" href="#E22:15_1"><span class="lab"><font color="Green" title="E27">A17</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">take </span><font color="Maroon" title="c3">X</font> =  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c2">f</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">X</font> is  <a href="ordinal1.html#V1" title="ORDINAL1:attr.1">epsilon-transitive</a>  &amp; <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c3">X</font> <a href="zf_colla.html#R2" title="ZF_COLLA:pred.2">are_epsilon-isomorphic</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E6:15"/>
<font color="Maroon" title="c3">X</font> is  <a href="ordinal1.html#V1" title="ORDINAL1:attr.1">epsilon-transitive</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="zf_colla.html#T10" target="_self" onmouseover="rs('zf_colla/T10')" onmouseout="rh()">Th10</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c3">X</font> <a href="zf_colla.html#R2" title="ZF_COLLA:pred.2">are_epsilon-isomorphic</a> </span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c2">f</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="zf_colla.html#D3" target="_self" onmouseover="rs('zf_colla/D3')" onmouseout="rh()">ZF_COLLA:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="zf_colla.html#R1" title="ZF_COLLA:pred.1">is_epsilon-isomorphism_of</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c3">X</font></span><br/>

<span class="kw">thus </span><a NAME="E7:15"/>
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font> &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E11">A1</font></span></a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="zf_colla.html#D2" target="_self" onmouseover="rs('zf_colla/D2')" onmouseout="rh()">ZF_COLLA:def 2</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; (  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font> holds <br/>(  ex <font color="Olive" title="b3">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b3">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="Olive" title="b3">Z</font> ) iff  ex <font color="Olive" title="b3">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">Z</font> &amp; <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">Z</font> ) ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E8:15"/>
<font color="Maroon" title="c2">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font> holds <br/>(  ex <font color="Olive" title="b3">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b3">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="Olive" title="b3">Z</font> ) iff  ex <font color="Olive" title="b3">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">Z</font> &amp; <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">Z</font> ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font> implies (  ex <font color="Olive" title="b1">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b1">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font> ) iff  ex <font color="Olive" title="b1">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Z</font> &amp; <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">Z</font> ) ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E9:15"/><span class="lab"><font color="Green" title="E15">A32</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font>
 <span class="kw">and </span><br/><a NAME="E10:15"/><span class="lab"><font color="Green" title="E16">A33</font></span>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b1">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font> ) iff  ex <font color="Olive" title="b1">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Z</font> &amp; <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">Z</font> ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">d2</font> = <font color="Maroon" title="c5">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:15"><span class="lab"><font color="Green" title="E16">A33</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E12:15"/>
(  ex <font color="Olive" title="b1">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b1">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font> ) implies  ex <font color="Olive" title="b1">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b1">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">Z</font> ) )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Z</font> &amp; <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">Z</font> ) implies  ex <font color="Olive" title="b1">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b1">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font> ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="15_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c7">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><a NAME="E2:15_2"/><span class="lab"><font color="Green" title="E17">A34</font></span>: 
( <font color="Maroon" title="c7">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b1">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">Z</font> )</span><br/>

<a NAME="E3:15_2"/><span class="lab"><font color="Green" title="E18">A35</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c6">d2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E12">A2</font></span></a></span>;<br/>
<a NAME="E4:15_2"/>
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c6">d2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E9:15"><span class="lab"><font color="Green" title="E15">A32</font></span></a>, <a class="txt" href="#E2:15_2"><span class="lab"><font color="Green" title="E17">A34</font></span></a>, <a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:15_2"/>
 ex <font color="Olive" title="b1">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b1">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">Z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_2"><span class="lab"><font color="Green" title="E18">A35</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">given </span><font color="Maroon" title="c7">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><a NAME="E14:15"/><span class="lab"><font color="Green" title="E17">A36</font></span>: 
( <font color="Maroon" title="c7">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">Z</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b1">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font> )</span><br/>

<a NAME="E15:15"/>
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c6">d2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E12">A2</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E17:15"/><span class="lab"><font color="Green" title="E18">A37</font></span>: 
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">and </span><br/><a NAME="E18:15"/><span class="lab"><font color="Green" title="E19">A38</font></span>: 
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">d2</font>
 <span class="kw">and </span><br/><a NAME="E19:15"/><span class="lab"><font color="Green" title="E20">A39</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:15"><span class="lab"><font color="Green" title="E17">A36</font></span></a>, <a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</a></span>;<br/>
<a NAME="E20:15"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="txt" href="#E9:15"><span class="lab"><font color="Green" title="E15">A32</font></span></a>, <a class="txt" href="#E17:15"><span class="lab"><font color="Green" title="E18">A37</font></span></a>, <a class="txt" href="#E19:15"><span class="lab"><font color="Green" title="E20">A39</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:15"/>
 ex <font color="Olive" title="b1">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b1">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:15"><span class="lab"><font color="Green" title="E19">A38</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
