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<b>let </b><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#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#M1" title="HIDDEN:mode.1">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/>

<b>assume </b><a NAME="E1:15"/><i><font color="Green" title="E11">A1</font></i>: 
<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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">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/>

<b>consider </b><font color="Maroon" title="c2">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:15"/><i><font color="Green" title="E12">A2</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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; ( 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#K9" title="RELAT_1:func.9">.:</a> <font color="Olive" title="b1">d</font> ) )
 <b>by </b><i><a class="ref" href="zf_colla.html#T9" target="_self" title="ZF_COLLA:th.9">Th9</a></i>;<br/>
<b>take </b><font color="Maroon" title="c3">X</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>thus </b><a NAME="E4:15"/>
<font color="Maroon" title="c3">X</font> is <a href="ordinal1.html#V1" title="ORDINAL1:attr.1">epsilon-transitive</a>
 <b>by </b><i><a class="txt" href="zf_colla.html#E3:15"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="zf_colla.html#T12" target="_self" title="ZF_COLLA:th.12">Th12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>take </b>
<font color="Maroon" title="c2">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_colla.html#D3" target="_self" title="ZF_COLLA:def.3">ZF_COLLA:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>thus </b><a NAME="E5:15"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="relat_1.html#K2" title="RELAT_1:func.2">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> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E3:15"><i><font color="Green" title="E12">A2</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_colla.html#D2" target="_self" title="ZF_COLLA:def.2">ZF_COLLA:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">set</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#M1" title="HIDDEN:mode.1">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#M1" title="HIDDEN:mode.1">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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">Z</font> ) ) ) )</span><br/>

<div><i><font color="Green" title="E13">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E2:15_1"/><i><font color="Green" title="E14">A4</font></i>: 
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</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="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> )
 <b>and </b><br/><a NAME="E3:15_1"/><i><font color="Green" title="E15">A5</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>reconsider </b><font color="Maroon" title="c6">d1</font> = <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c7">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> <b>by </b><i><a class="txt" href="zf_colla.html#E3:15"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="zf_colla.html#E2:15_1"><i><font color="Green" title="E14">A4</font></i></a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b> 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="hidden.html#R2" title="HIDDEN: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="hidden.html#R2" title="HIDDEN: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/><b>consider </b><font color="Maroon" title="c8">A1</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E6:15_1"/><i><font color="Green" title="E16">A6</font></i>: 
<font color="Maroon" title="c6">d1</font> <a href="hidden.html#R2" title="HIDDEN: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">A1</font>
 <b>by </b><i><a class="ref" href="zf_colla.html#T5" target="_self" title="ZF_COLLA:th.5">Th5</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">A2</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E8:15_1"/><i><font color="Green" title="E17">A7</font></i>: 
<font color="Maroon" title="c7">d2</font> <a href="hidden.html#R2" title="HIDDEN: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">A2</font>
 <b>by </b><i><a class="ref" href="zf_colla.html#T5" target="_self" title="ZF_COLLA:th.5">Th5</a></i>;<br/><a NAME="E9:15_1"/>
( <font color="Maroon" title="c8">A1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c9">A2</font> or <font color="Maroon" title="c9">A2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">A1</font> )
 ;<br/><a NAME="E10:15_1"/><b>then </b>
(  <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">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="c9">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="c9">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="c8">A1</font> )
 <b>by </b><i><a class="ref" href="zf_colla.html#T4" target="_self" title="ZF_COLLA:th.4">Th4</a></i>;<br/><a NAME="E11:15_1"/><b>then </b><i><font color="Green" title="E18">A8</font></i>: 
 ex <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>]
 <b>by </b><i><a class="txt" href="zf_colla.html#E2:15_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="zf_colla.html#E3:15_1"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="zf_colla.html#E6:15_1"><i><font color="Green" title="E16">A6</font></i></a>, <a class="txt" href="zf_colla.html#E8:15_1"><i><font color="Green" title="E17">A7</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c10">A</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E13:15_1"/><i><font color="Green" title="E19">A9</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">A</font>] &amp; ( for <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B</font>] holds <br/><font color="Maroon" title="c10">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">B</font> ) )
 <b>from </b><i><a class="ref" href="ordinal1.html#S1" title="ORDINAL1:sch.1">ORDINAL1:sch 1</a>(<a class="txt" href="zf_colla.html#E11:15_1"><i><font color="Green" title="E18">A8</font></i></a>);<br/></i><b>consider </b><font color="Maroon" title="c11">d1</font>, <font color="Maroon" title="c12">d2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font><b> such that </b><br/><a NAME="E15:15_1"/><i><font color="Green" title="E20">A10</font></i>: 
( <font color="Maroon" title="c11">d1</font> <a href="hidden.html#R2" title="HIDDEN: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="c10">A</font> &amp; <font color="Maroon" title="c12">d2</font> <a href="hidden.html#R2" title="HIDDEN: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="c10">A</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">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="c12">d2</font> &amp; <font color="Maroon" title="c11">d1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">d2</font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E13:15_1"><i><font color="Green" title="E19">A9</font></i></a></i>;<br/><a NAME="E16:15_1"/><i><font color="Green" title="E21">A11</font></i>: 
( <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c11">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="c12">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c12">d2</font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E3:15"><i><font color="Green" title="E12">A2</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c13">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font><b> such that </b><br/><a NAME="E18:15_1"/><i><font color="Green" title="E22">A12</font></i>: 
( ( <font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">d1</font> &amp; not <font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">d2</font> ) or ( <font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">d2</font> &amp; not <font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">d1</font> ) )
 <b>by </b><i><a class="txt" href="zf_colla.html#E1:15"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="zf_colla.html#E15:15_1"><i><font color="Green" title="E20">A10</font></i></a>, <a class="ref" href="zf_colla.html#T13" target="_self" title="ZF_COLLA:th.13">Th13</a></i>;<br/><div><i><font color="Green" title="E23">A13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_1"/><i><font color="Green" title="E24">A14</font></i>: 
( <font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">d1</font> &amp; not <font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">d2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:15_1_1"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c11">d1</font>
 <b>by </b><i><a class="txt" href="zf_colla.html#E3:15"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><b>then consider </b><font color="Maroon" title="c14">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:15_1_1"/><i><font color="Green" title="E25">A15</font></i>: 
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">d2</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="c13">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> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E15:15_1"><i><font color="Green" title="E20">A10</font></i></a>, <a class="txt" href="zf_colla.html#E16:15_1"><i><font color="Green" title="E21">A11</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><b>reconsider </b><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> <b>by </b><i><a class="txt" href="zf_colla.html#E3:15"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="zf_colla.html#E4:15_1_1"><i><font color="Green" title="E25">A15</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c16">A1</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E7:15_1_1"/><i><font color="Green" title="E26">A16</font></i>: 
( <font color="Maroon" title="c16">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> &amp; <font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN: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">A1</font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E15:15_1"><i><font color="Green" title="E20">A10</font></i></a>, <a class="txt" href="zf_colla.html#E1:15_1_1"><i><font color="Green" title="E24">A14</font></i></a>, <a class="ref" href="zf_colla.html#T6" target="_self" title="ZF_COLLA:th.6">Th6</a></i>;<br/><b>consider </b><font color="Maroon" title="c17">A2</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E9:15_1_1"/><i><font color="Green" title="E27">A17</font></i>: 
( <font color="Maroon" title="c17">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> &amp; <font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN: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="c17">A2</font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E15:15_1"><i><font color="Green" title="E20">A10</font></i></a>, <a class="txt" href="zf_colla.html#E4:15_1_1"><i><font color="Green" title="E25">A15</font></i></a>, <a class="ref" href="zf_colla.html#T6" target="_self" title="ZF_COLLA:th.6">Th6</a></i>;<br/><a NAME="E10:15_1_1"/>
( <font color="Maroon" title="c16">A1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c17">A2</font> or <font color="Maroon" title="c17">A2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c16">A1</font> )
 ;<br/><a NAME="E11:15_1_1"/><b>then </b>
(  <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">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="c17">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="c17">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="c16">A1</font> )
 <b>by </b><i><a class="ref" href="zf_colla.html#T4" target="_self" title="ZF_COLLA:th.4">Th4</a></i>;<br/><b>then consider </b><font color="Maroon" title="c18">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E13:15_1_1"/><i><font color="Green" title="E28">A18</font></i>: 
( <font color="Maroon" title="c18">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> &amp; <font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN: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="c18">B</font> &amp; <font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN: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="c18">B</font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E7:15_1_1"><i><font color="Green" title="E26">A16</font></i></a>, <a class="txt" href="zf_colla.html#E9:15_1_1"><i><font color="Green" title="E27">A17</font></i></a></i>;<br/><b>thus </b><a NAME="E14:15_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="zf_colla.html#E13:15_1"><i><font color="Green" title="E19">A9</font></i></a>, <a class="txt" href="zf_colla.html#E1:15_1_1"><i><font color="Green" title="E24">A14</font></i></a>, <a class="txt" href="zf_colla.html#E4:15_1_1"><i><font color="Green" title="E25">A15</font></i></a>, <a class="txt" href="zf_colla.html#E13:15_1_1"><i><font color="Green" title="E28">A18</font></i></a>, <a class="ref" href="ordinal1.html#T7" title="ORDINAL1:th.7">ORDINAL1:7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_2"/><i><font color="Green" title="E24">A19</font></i>: 
( <font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">d2</font> &amp; not <font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">d1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:15_1_2"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c12">d2</font>
 <b>by </b><i><a class="txt" href="zf_colla.html#E3:15"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><b>then consider </b><font color="Maroon" title="c14">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:15_1_2"/><i><font color="Green" title="E25">A20</font></i>: 
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">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="c13">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> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E15:15_1"><i><font color="Green" title="E20">A10</font></i></a>, <a class="txt" href="zf_colla.html#E16:15_1"><i><font color="Green" title="E21">A11</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><b>reconsider </b><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> <b>by </b><i><a class="txt" href="zf_colla.html#E3:15"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="zf_colla.html#E4:15_1_2"><i><font color="Green" title="E25">A20</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c16">A1</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E7:15_1_2"/><i><font color="Green" title="E26">A21</font></i>: 
( <font color="Maroon" title="c16">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> &amp; <font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN: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">A1</font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E15:15_1"><i><font color="Green" title="E20">A10</font></i></a>, <a class="txt" href="zf_colla.html#E1:15_1_2"><i><font color="Green" title="E24">A19</font></i></a>, <a class="ref" href="zf_colla.html#T6" target="_self" title="ZF_COLLA:th.6">Th6</a></i>;<br/><b>consider </b><font color="Maroon" title="c17">A2</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E9:15_1_2"/><i><font color="Green" title="E27">A22</font></i>: 
( <font color="Maroon" title="c17">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> &amp; <font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN: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="c17">A2</font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E15:15_1"><i><font color="Green" title="E20">A10</font></i></a>, <a class="txt" href="zf_colla.html#E4:15_1_2"><i><font color="Green" title="E25">A20</font></i></a>, <a class="ref" href="zf_colla.html#T6" target="_self" title="ZF_COLLA:th.6">Th6</a></i>;<br/><a NAME="E10:15_1_2"/>
( <font color="Maroon" title="c16">A1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c17">A2</font> or <font color="Maroon" title="c17">A2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c16">A1</font> )
 ;<br/><a NAME="E11:15_1_2"/><b>then </b>
(  <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">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="c17">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="c17">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="c16">A1</font> )
 <b>by </b><i><a class="ref" href="zf_colla.html#T4" target="_self" title="ZF_COLLA:th.4">Th4</a></i>;<br/><b>then consider </b><font color="Maroon" title="c18">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E13:15_1_2"/><i><font color="Green" title="E28">A23</font></i>: 
( <font color="Maroon" title="c18">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> &amp; <font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN: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="c18">B</font> &amp; <font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN: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="c18">B</font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E7:15_1_2"><i><font color="Green" title="E26">A21</font></i></a>, <a class="txt" href="zf_colla.html#E9:15_1_2"><i><font color="Green" title="E27">A22</font></i></a></i>;<br/><b>thus </b><a NAME="E14:15_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="zf_colla.html#E13:15_1"><i><font color="Green" title="E19">A9</font></i></a>, <a class="txt" href="zf_colla.html#E1:15_1_2"><i><font color="Green" title="E24">A19</font></i></a>, <a class="txt" href="zf_colla.html#E4:15_1_2"><i><font color="Green" title="E25">A20</font></i></a>, <a class="txt" href="zf_colla.html#E13:15_1_2"><i><font color="Green" title="E28">A23</font></i></a>, <a class="ref" href="ordinal1.html#T7" title="ORDINAL1:th.7">ORDINAL1:7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E21:15_1"/>
contradiction
 <b>by </b><i><a class="txt" href="zf_colla.html#E18:15_1"><i><font color="Green" title="E22">A12</font></i></a>, <a class="txt" href="zf_colla.html#E19:15_1"><i><font color="Green" title="E23">A13</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:15"/>
<font color="Maroon" title="c2">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">set</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#M1" title="HIDDEN:mode.1">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#M1" title="HIDDEN:mode.1">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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">Z</font> ) )</span><br/>

<b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#M1" title="HIDDEN:mode.1">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#M1" title="HIDDEN:mode.1">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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font> ) ) )</span><br/>



<b>assume </b><a NAME="E8:15"/><i><font color="Green" title="E14">A24</font></i>: 
( <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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">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#M1" title="HIDDEN:mode.1">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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font> ) )</span><br/>

<b>then reconsider </b><font color="Maroon" title="c6">d1</font> = <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c7">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> ;<br/>
<b>thus </b><a NAME="E10:15"/>
(  ex <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">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#M1" title="HIDDEN:mode.1">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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">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="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#M1" title="HIDDEN:mode.1">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" onclick="hs2(this)" href="javascript:()" title="15_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c8">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E2:15_2"/><i><font color="Green" title="E15">A25</font></i>: 
( <font color="Maroon" title="c8">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="c8">Z</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font> )</span><br/>

<a NAME="E3:15_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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c7">d2</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="c6">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c6">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="c7">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c7">d2</font> )
 
<b>by </b><i><a class="txt" href="zf_colla.html#E3:15"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="zf_colla.html#E8:15"><i><font color="Green" title="E14">A24</font></i></a>, <a class="txt" href="zf_colla.html#E2:15_2"><i><font color="Green" title="E15">A25</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E4:15_2"/>
 ex <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>given </b><font color="Maroon" title="c8">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E12:15"/><i><font color="Green" title="E15">A26</font></i>: 
( <font color="Maroon" title="c8">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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">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="E13:15"/>
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c7">d2</font>
 
<b>by </b><i><a class="txt" href="zf_colla.html#E3:15"><i><font color="Green" title="E12">A2</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E15:15"/><i><font color="Green" title="E16">A27</font></i>: 
( <font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">d2</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="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="c9">c</font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E12:15"><i><font color="Green" title="E15">A26</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E16:15"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">c</font>
 
<b>by </b><i><a class="txt" href="zf_colla.html#E3:15"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="zf_colla.html#E6:15"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="zf_colla.html#E8:15"><i><font color="Green" title="E14">A24</font></i></a>, <a class="txt" href="zf_colla.html#E15:15"><i><font color="Green" title="E16">A27</font></i></a></i>;<br/>
<b>hence </b><a NAME="E17:15"/>
 ex <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">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> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E15:15"><i><font color="Green" title="E16">A27</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
