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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"> 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>  ex <font color="Olive" title="b2">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <font color="Olive" title="b1">d</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="Olive" title="b2">A</font></span><br/><b>let </b><font color="Maroon" title="c2">d</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <font color="Maroon" title="c2">d</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="Olive" title="b1">A</font></span><br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  holds not $1 <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="Olive" title="b1">A</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> &amp; $1 <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="Olive" title="b1">A</font> &amp; ( for <font color="Olive" title="b2">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st $1 <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="Olive" title="b2">B</font> holds <br/><font color="Olive" title="b1">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b2">B</font> ) );<br/>
<b>consider </b><font color="Maroon" title="c3">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:6"/><i><font color="Green" title="E5">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> iff ( <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="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<b>assume </b><a NAME="E3:6"/><i><font color="Green" title="E6">A2</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  holds not <font color="Maroon" title="c2">d</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="Olive" title="b1">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E2:6_1"/><i><font color="Green" title="E7">A3</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>consider </b><font color="Maroon" title="c5">m</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:6_1"/><i><font color="Green" title="E8">A4</font></i>: 
( <font color="Maroon" title="c5">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> or  not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">m</font> ) ) )
 <b>by </b><i><a class="txt" href="zf_colla.html#E2:6_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="tarski.html#T7" title="TARSKI:th.7">TARSKI:7</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">m</font> = <font color="Maroon" title="c5">m</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#E2:6"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="zf_colla.html#E4:6_1"><i><font color="Green" title="E8">A4</font></i></a></i>;<br/><div><i><font color="Green" title="E9">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font>, <font color="Maroon" title="c9">z</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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">m</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c1">E</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">z</font>] implies <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">z</font> )</span><br/><b>assume </b><a NAME="E1:6_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">m</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</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"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">z</font>] implies <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">z</font> )</span><br/><b>assume </b><a NAME="E2:6_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">z</font>] implies <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">z</font> )</span><br/><b>then consider </b><font color="Maroon" title="c10">A1</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E4:6_1_1"/><i><font color="Green" title="E10">A6</font></i>: 
( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">A1</font> &amp; <font color="Maroon" title="c7">x</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">A1</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" title="c7">x</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="Olive" title="b1">B</font> holds <br/><font color="Maroon" title="c10">A1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">B</font> ) )
 ;<br/><b>assume </b><a NAME="E5:6_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">z</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">z</font></span><br/><b>then consider </b><font color="Maroon" title="c11">A2</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E7:6_1_1"/><i><font color="Green" title="E11">A7</font></i>: 
( <font color="Maroon" title="c9">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">A2</font> &amp; <font color="Maroon" title="c7">x</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="c11">A2</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" title="c7">x</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="Olive" title="b1">B</font> holds <br/><font color="Maroon" title="c11">A2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">B</font> ) )
 ;<br/><a NAME="E8:6_1_1"/>
( <font color="Maroon" title="c10">A1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c11">A2</font> &amp; <font color="Maroon" title="c11">A2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c10">A1</font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E4:6_1_1"><i><font color="Green" title="E10">A6</font></i></a>, <a class="txt" href="zf_colla.html#E7:6_1_1"><i><font color="Green" title="E11">A7</font></i></a></i>;<br/><b>hence </b><a NAME="E9:6_1_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">z</font>
 <b>by </b><i><a class="txt" href="zf_colla.html#E4:6_1_1"><i><font color="Green" title="E10">A6</font></i></a>, <a class="txt" href="zf_colla.html#E7:6_1_1"><i><font color="Green" title="E11">A7</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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><i><font color="Green" title="E10">A8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">x</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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">m</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c1">E</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b><font color="Maroon" title="c7">x</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;<br/><b>assume </b><a NAME="E1:6_1_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">m</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</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">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><a NAME="E2:6_1_2"/><b>then </b><i><font color="Green" title="E11">A9</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">m</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:6_1_2"/><b>then </b>
not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="zf_colla.html#E4:6_1"><i><font color="Green" title="E8">A4</font></i></a></i>;<br/><a NAME="E4:6_1_2"/><b>then </b><i><font color="Green" title="E12">A10</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>3</sub></font>[<font color="Olive" title="b1">A</font>]
 <b>by </b><i><a class="txt" href="zf_colla.html#E2:6"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="zf_colla.html#E2:6_1_2"><i><font color="Green" title="E11">A9</font></i></a></i>;<br/><a NAME="E5:6_1_2"/>
 ex <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">A</font>] &amp; ( for <font color="Olive" title="b2">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b2">B</font>] holds <br/><font color="Olive" title="b1">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b2">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#E4:6_1_2"><i><font color="Green" title="E12">A10</font></i></a>);<br/></i><b>hence </b><a NAME="E6:6_1_2"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</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>consider </b><font color="Maroon" title="c7">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E9:6_1"/><i><font color="Green" title="E11">A11</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c1">E</font> &amp; ( for <font color="Olive" title="b1">x</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="c6">m</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c1">E</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2" title="FUNCT_1:sch.2">FUNCT_1:sch 2</a>(<a class="txt" href="zf_colla.html#E6:6_1"><i><font color="Green" title="E9">A5</font></i></a>, <a class="txt" href="zf_colla.html#E7:6_1"><i><font color="Green" title="E10">A8</font></i></a>);<br/></i><a NAME="E10:6_1"/>
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">f</font> holds <br/><font color="Olive" title="b1">y</font> is   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">y</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="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">f</font> implies <font color="Maroon" title="c8">y</font> is   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> )</span><br/>

<b>assume </b><a NAME="E1:6_1_3"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">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="c8">y</font> is   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a></span><br/>

<b>then consider </b><font color="Maroon" title="c9">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:6_1_3"/><i><font color="Green" title="E12">A12</font></i>: 
( <font color="Maroon" title="c9">x</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="c7">f</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:6_1_3"/>
 ex <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> &amp; <font color="Maroon" title="c9">x</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="Olive" title="b1">A</font> &amp; ( for <font color="Olive" title="b2">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon" title="c9">x</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="Olive" title="b2">B</font> holds <br/><font color="Olive" title="b1">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b2">B</font> ) )
 
<b>by </b><i><a class="txt" href="zf_colla.html#E9:6_1"><i><font color="Green" title="E11">A11</font></i></a>, <a class="txt" href="zf_colla.html#E3:6_1_3"><i><font color="Green" title="E12">A12</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:6_1_3"/>
<font color="Maroon" title="c8">y</font> is   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>
 <b>by </b><i><a class="txt" href="zf_colla.html#E3:6_1_3"><i><font color="Green" title="E12">A12</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>then consider </b><font color="Maroon" title="c8">A</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E12:6_1"/><i><font color="Green" title="E12">A13</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">A</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T36" title="ORDINAL1:th.36">ORDINAL1:36</a></i>;<br/><a NAME="E13:6_1"/>
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>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">m</font> holds <br/> ex <font color="Olive" title="b2">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font> &amp; <font color="Olive" title="b1">d</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="Olive" title="b2">B</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">d</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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"> ( <font color="Maroon" title="c9">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">m</font> implies  ex <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font> &amp; <font color="Maroon" title="c9">d</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="Olive" title="b1">B</font> ) )</span><br/>

<b>assume </b><a NAME="E1:6_1_4"/>
<font color="Maroon" title="c9">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">m</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">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font> &amp; <font color="Maroon" title="c9">d</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="Olive" title="b1">B</font> )</span><br/>

<a NAME="E2:6_1_4"/><b>then </b><i><font color="Green" title="E13">A14</font></i>: 
<font color="Maroon" title="c9">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">m</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c1">E</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E4:6_1_4"/><i><font color="Green" title="E14">A15</font></i>: 
( <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c9">d</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">B</font> &amp; ( for <font color="Olive" title="b1">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon" title="c9">d</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="Olive" title="b1">C</font> holds <br/><font color="Maroon" title="c10">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">C</font> ) )
 <b>by </b><i><a class="txt" href="zf_colla.html#E9:6_1"><i><font color="Green" title="E11">A11</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c10">B</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font> &amp; <font color="Maroon" title="c9">d</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">B</font> )</span><br/>

<a NAME="E5:6_1_4"/>
<font color="Maroon" title="c10">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">f</font>
 
<b>by </b><i><a class="txt" href="zf_colla.html#E9:6_1"><i><font color="Green" title="E11">A11</font></i></a>, <a class="txt" href="zf_colla.html#E2:6_1_4"><i><font color="Green" title="E13">A14</font></i></a>, <a class="txt" href="zf_colla.html#E4:6_1_4"><i><font color="Green" title="E14">A15</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:6_1_4"/>
( <font color="Maroon" title="c10">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font> &amp; <font color="Maroon" title="c9">d</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">B</font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E12:6_1"><i><font color="Green" title="E12">A13</font></i></a>, <a class="txt" href="zf_colla.html#E4:6_1_4"><i><font color="Green" title="E14">A15</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><a NAME="E14:6_1"/><b>then </b>
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">d'</font> where <font color="Olive" title="b1">d'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> : for <font color="Olive" title="b2">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font>  st <font color="Olive" title="b2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">d'</font> holds <br/> ex <font color="Olive" title="b3">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font> &amp; <font color="Olive" title="b2">d</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="Olive" title="b3">B</font> ) </span>}</span> 
 ;<br/><a NAME="E15:6_1"/><b>then </b>
( <font color="Maroon" title="c6">m</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">A</font> &amp; ( for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  holds not <font color="Maroon" title="c6">m</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="Olive" title="b1">A</font> ) )
 <b>by </b><i><a class="txt" href="zf_colla.html#E2:6"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="zf_colla.html#E4:6_1"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="zf_colla.html#T1" target="_self" title="ZF_COLLA:th.1">Th1</a></i>;<br/><b>hence </b><a NAME="E16:6_1"/>
contradiction
 ; <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>
<a NAME="E5:6"/><b>then </b>
not <font color="Maroon" title="c2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 
;<br/>
<b>hence </b><a NAME="E6:6"/>
contradiction
 <b>by </b><i><a class="txt" href="zf_colla.html#E2:6"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="zf_colla.html#E3:6"><i><font color="Green" title="E6">A2</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>
