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<b>defpred </b><font color="Maroon">S<sub>1</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/>( <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; <font color="Maroon">P<sub>1</sub></font>[$1,<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">P<sub>1</sub></font>[$1,<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/>
<a NAME="E1:70_1"/><i><font color="Green" title="E36">A2</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</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">F<sub>1</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</font>] holds <br/><font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">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="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">z</font>] implies <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font> )</span><br/>





<b>assume </b><a NAME="E1:70_1_1"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>] or  not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">z</font>] or <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font> )</span><br/>

<b>given </b><font color="Maroon" title="c4">A1</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><a NAME="E3:70_1_1"/><i><font color="Green" title="E37">A3</font></i>: 
( <font color="Maroon" title="c4">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c4">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">P<sub>1</sub></font>[<font color="Maroon" title="c1">x</font>,<font color="Olive" title="b1">B</font>] holds <br/><font color="Maroon" title="c4">A1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">B</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">z</font>] or <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font> )</span><br/>

<b>given </b><font color="Maroon" title="c5">A2</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><a NAME="E5:70_1_1"/><i><font color="Green" title="E38">A4</font></i>: 
( <font color="Maroon" title="c5">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c5">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">P<sub>1</sub></font>[<font color="Maroon" title="c1">x</font>,<font color="Olive" title="b1">B</font>] holds <br/><font color="Maroon" title="c5">A2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">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="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font></span><br/>

<a NAME="E6:70_1_1"/>
( <font color="Maroon" title="c4">A1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">A2</font> &amp; <font color="Maroon" title="c5">A2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">A1</font> )
 
<b>by </b><i><a class="txt" href="#E3:70_1_1"><i><font color="Green" title="E37">A3</font></i></a>, <a class="txt" href="#E5:70_1_1"><i><font color="Green" title="E38">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:70_1_1"/>
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font>
 <b>by </b><i><a class="txt" href="#E3:70_1_1"><i><font color="Green" title="E37">A3</font></i></a>, <a class="txt" href="#E5:70_1_1"><i><font color="Green" title="E38">A4</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>
<a NAME="E2:70_1"/><i><font color="Green" title="E37">A5</font></i>: 
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">F<sub>1</sub></font>() holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">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="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></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>1</sub></font>[<font color="Maroon" title="c1">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<b>assume </b><a NAME="E1:70_1_2"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></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>1</sub></font>[<font color="Maroon" title="c1">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<b>then reconsider </b><font color="Maroon" title="c2">d</font> = <font color="Maroon" title="c1">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">d</font>,$1];<br/>
<a NAME="E3:70_1_2"/><i><font color="Green" title="E38">A6</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>2</sub></font>[<font color="Olive" title="b1">A</font>]
 
<b>by </b><i><a class="txt" href="#E1:70"><i><font color="Green" title="E35">A1</font></i></a></i>;<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="E5:70_1_2"/><i><font color="Green" title="E39">A7</font></i>: 
( <font color="Maroon">S<sub>2</sub></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">S<sub>2</sub></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>from </b><i><a class="ref" href="ordinal1.html#S1" target="_self" title="ORDINAL1:sch.1">ORDINAL1:sch 1</a>(<a class="txt" href="#E3:70_1_2"><i><font color="Green" title="E38">A6</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c4">y</font> = <font color="Maroon" title="c3">A</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon" title="c4">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>1</sub></font>[<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c4">y</font>]</span><br/>

<b>take </b>
<font color="Maroon" title="c3">A</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">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">x</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="c1">x</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> ) )</span><br/>

<b>thus </b><a NAME="E7:70_1_2"/>
( <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">x</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="c1">x</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="#E5:70_1_2"><i><font color="Green" title="E39">A7</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>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="E4:70_1"/><i><font color="Green" title="E38">A8</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>1</sub></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">F<sub>1</sub></font>() holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c1">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="#E1:70_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="#E2:70_1"><i><font color="Green" title="E37">A5</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c1">F</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <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>1</sub></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">F<sub>1</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> ) ) ) )</span><br/>

<b>thus </b><a NAME="E5:70_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>1</sub></font>()
 <b>by </b><i><a class="txt" href="#E4:70_1"><i><font color="Green" title="E38">A8</font></i></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">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</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> ) )</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">F<sub>1</sub></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 <br/>( <font color="Olive" title="b1">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="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">P<sub>1</sub></font>[<font color="Maroon" title="c2">d</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> ) )</span><br/>

<b>thus </b><a NAME="E6:70_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">d</font>,<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>]
 <b>by </b><i><a class="txt" href="#E4:70_1"><i><font color="Green" title="E38">A8</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>
