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<a NAME="E1:7_1"/><i><font color="Green" title="E7">A2</font></i>: 
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">F<sub>2</sub></font>()  ex <font color="Olive" title="b2">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">d</font>,<font color="Olive" title="b2">A</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">d</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>();<br/>

<b>consider </b><font color="Maroon" title="c2">a</font> being    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E2:7_1_1"/><i><font color="Green" title="E8">A3</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">d</font>,<font color="Maroon" title="c2">a</font>]
 <b>by </b><i><a class="txt" href="zf_refle.html#E1:7"><i><font color="Green" title="E6">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">a</font> = <font color="Maroon" title="c2">a</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/>
<b>take </b>
<font color="Maroon" title="c3">a</font>
;<br/>

<b>thus </b><a NAME="E4:7_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">d</font>,<font color="Maroon" title="c3">a</font>]
 <b>by </b><i><a class="txt" href="zf_refle.html#E2:7_1_1"><i><font color="Green" title="E8">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c1">F</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:7_1"/><i><font color="Green" title="E8">A4</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>and </b><br/><a NAME="E4:7_1"/><i><font color="Green" title="E9">A5</font></i>: 
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">F<sub>2</sub></font>()  ex <font color="Olive" title="b2">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">d</font>,<font color="Olive" title="b2">A</font>] &amp; ( for <font color="Olive" title="b3">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">d</font>,<font color="Olive" title="b3">B</font>] holds <br/><font color="Olive" title="b2">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b3">B</font> ) )
 <b>from </b><i><a class="ref" href="ordinal1.html#S6" title="ORDINAL1:sch.6">ORDINAL1:sch 6</a>(<a class="txt" href="zf_refle.html#E1:7_1"><i><font color="Green" title="E7">A2</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c1">F</font>
;<br/>

<b>thus </b><a NAME="E5:7_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i><a class="txt" href="zf_refle.html#E3:7_1"><i><font color="Green" title="E8">A4</font></i></a></i>;<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">F<sub>2</sub></font>();<br/>

<b>consider </b><font color="Maroon" title="c3">A</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E7:7_1"/><i><font color="Green" title="E10">A6</font></i>: 
( <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">d</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">d</font>,<font color="Maroon" title="c3">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">P<sub>1</sub></font>[<font color="Maroon" title="c2">d</font>,<font color="Olive" title="b1">B</font>] holds <br/><font color="Maroon" title="c3">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">B</font> ) )
 <b>by </b><i><a class="txt" href="zf_refle.html#E4:7_1"><i><font color="Green" title="E9">A5</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">a</font> being    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E9:7_1"/><i><font color="Green" title="E11">A7</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">d</font>,<font color="Maroon" title="c4">a</font>]
 <b>by </b><i><a class="txt" href="zf_refle.html#E1:7"><i><font color="Green" title="E6">A1</font></i></a></i>;<br/>
<a NAME="E10:7_1"/>
( <font color="Maroon" title="c3">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E7:7_1"><i><font color="Green" title="E10">A6</font></i></a>, <a class="txt" href="zf_refle.html#E9:7_1"><i><font color="Green" title="E11">A7</font></i></a>, <a class="ref" href="ordinal4.html#D2" title="ORDINAL4:def.2">ORDINAL4:def 2</a></i>;<br/>
<a NAME="E11:7_1"/><b>then </b>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="classes1.html#D1" title="CLASSES1:def.1">CLASSES1:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c3">A</font> as    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="ordinal4.html#D2" title="ORDINAL4:def.2">ORDINAL4:def 2</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">a</font>
;<br/>

<b>thus </b><a NAME="E13:7_1"/>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">d</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">d</font>,<font color="Maroon" title="c5">a</font>] &amp; ( for <font color="Olive" title="b1">b</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">d</font>,<font color="Olive" title="b1">b</font>] holds <br/><font color="Maroon" title="c5">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">b</font> ) )
 <b>by </b><i><a class="txt" href="zf_refle.html#E7:7_1"><i><font color="Green" title="E10">A6</font></i></a></i>;<br/>


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