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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_1_1"/>
( <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or ( <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_1"><b>suppose </b></a><a NAME="E1:12_1_1_1"/><i><font color="Green" title="E6">A2</font></i>: 
<font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() st <br/>for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">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">P<sub>1</sub></font>[<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font>]</span><br/><div class="add"><div><i><font color="Green" title="E7">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1_1"><b>now </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"> not <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>()</span><br/><b>assume </b><a NAME="E1:12_1_1_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"> contradiction</span><br/><a NAME="E2:12_1_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">x</font>,<font color="Olive" title="b1">y</font>] )
 <b>by </b><i><a class="txt" href="funct_2.html#E1:12"><i><font color="Green" title="E5">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E3:12_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="funct_2.html#E1:12_1_1_1"><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><b>end;</b></div><b>consider </b><font color="Maroon" title="c1">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>();<br/><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"> 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">P<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>]</span><br/><b>thus </b><a NAME="E4:12_1_1_1"/>
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">P<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>by </b><i><a class="txt" href="funct_2.html#E2:12_1_1_1"><i><font color="Green" title="E7">A3</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_2"><b>suppose </b></a><a NAME="E1:12_1_1_2"/><i><font color="Green" title="E6">A4</font></i>: 
<font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() st <br/>for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">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">P<sub>1</sub></font>[<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font>]</span><br/><div class="add"><b>consider </b><font color="Maroon" title="c1">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>();<br/><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"> 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">P<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>]</span><br/><b>thus </b><a NAME="E3:12_1_1_2"/>
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">P<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>by </b><i><a class="txt" href="funct_2.html#E1:12_1_1_2"><i><font color="Green" title="E6">A4</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_3"><b>suppose </b></a><a NAME="E1:12_1_1_3"/>
( <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() st <br/>for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">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">P<sub>1</sub></font>[<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font>]</span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c1">X</font> = <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon" title="c2">Y</font> = <font color="Maroon">F<sub>2</sub></font>() as  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>  ;<br/><b>set </b><font color="Maroon" title="c3">M</font> = <span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <font color="Olive" title="b2">ff</font> where <font color="Olive" title="b2">ff</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">tt</font>,<font color="Olive" title="b2">ff</font>] </span>}</span>  where <font color="Olive" title="b1">tt</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : verum </span>}</span> ;<br/><b>consider </b><font color="Maroon" title="c4">x0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/><a NAME="E4:12_1_1_3"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">ff</font> where <font color="Olive" title="b1">ff</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">x0</font>,<font color="Olive" title="b1">ff</font>] </span>}</span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <font color="Olive" title="b2">ff</font> where <font color="Olive" title="b2">ff</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">tt</font>,<font color="Olive" title="b2">ff</font>] </span>}</span>  where <font color="Olive" title="b1">tt</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : verum </span>}</span> 
 ;<br/><b>then reconsider </b><font color="Maroon" title="c5">M</font> = <span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <font color="Olive" title="b2">ff</font> where <font color="Olive" title="b2">ff</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">tt</font>,<font color="Olive" title="b2">ff</font>] </span>}</span>  where <font color="Olive" title="b1">tt</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : verum </span>}</span>  as  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>  ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">M</font> implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/><b>assume </b><a NAME="E1:12_1_1_3_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><b>then consider </b><font color="Maroon" title="c7">t</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:12_1_1_3_1"/><i><font color="Green" title="E6">A5</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">ff</font> where <font color="Olive" title="b1">ff</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c7">t</font>,<font color="Olive" title="b1">ff</font>] </span>}</span> 
 ;<br/><b>consider </b><font color="Maroon" title="c8">ff</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:12_1_1_3_1"/><i><font color="Green" title="E7">A6</font></i>: 
<font color="Maroon" title="c8">ff</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 <b>and </b><br/><a NAME="E6:12_1_1_3_1"/><i><font color="Green" title="E8">A7</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c7">t</font>,<font color="Maroon" title="c8">ff</font>]
 <b>by </b><i><a class="txt" href="funct_2.html#E1:12"><i><font color="Green" title="E5">A1</font></i></a></i>;<br/><a NAME="E7:12_1_1_3_1"/>
<font color="Maroon" title="c8">ff</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="funct_2.html#E3:12_1_1_3_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="funct_2.html#E5:12_1_1_3_1"><i><font color="Green" title="E7">A6</font></i></a>, <a class="txt" href="funct_2.html#E6:12_1_1_3_1"><i><font color="Green" title="E8">A7</font></i></a></i>;<br/><b>hence </b><a NAME="E8:12_1_1_3_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <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="c6">C</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E8:12_1_1_3"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">M</font>
 <b>and </b><br/><a NAME="E9:12_1_1_3"/><i><font color="Green" title="E6">A8</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" title="c5">M</font> holds <br/><font color="Maroon" title="c6">C</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="b1">x</font>
 <b>by </b><i><a class="ref" href="wellord2.html#T28" title="WELLORD2:th.28">WELLORD2:28</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">ff</font> where <font color="Olive" title="b1">ff</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> : <font color="Maroon">P<sub>1</sub></font>[$1,<font color="Olive" title="b1">ff</font>] </span>}</span> ;<br/><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="E11:12_1_1_3"/><i><font color="Green" title="E7">A9</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="c1">X</font>
 <b>and </b><br/><a NAME="E12:12_1_1_3"/><i><font color="Green" title="E8">A10</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" title="c1">X</font> holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></i><a NAME="E13:12_1_1_3"/>
 <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="c2">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c8">x</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> or <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> )</span><br/>

<b>assume </b><a NAME="E1:12_1_1_3_2"/>
<font color="Maroon" title="c8">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font></span><br/>

<b>then consider </b><font color="Maroon" title="c9">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:12_1_1_3_2"/><i><font color="Green" title="E9">A11</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>and </b><br/><a NAME="E4:12_1_1_3_2"/><i><font color="Green" title="E10">A12</font></i>: 
<font color="Maroon" title="c8">x</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">y</font>
 <b>by </b><i><a class="txt" href="funct_2.html#E11:12_1_1_3"><i><font color="Green" title="E7">A9</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">y0</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">ff</font> where <font color="Olive" title="b1">ff</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c9">y</font>,<font color="Olive" title="b1">ff</font>] </span>}</span> ;<br/>
<a NAME="E5:12_1_1_3_2"/><i><font color="Green" title="E11">A13</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">ff</font> where <font color="Olive" title="b1">ff</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c9">y</font>,<font color="Olive" title="b1">ff</font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="funct_2.html#E12:12_1_1_3"><i><font color="Green" title="E8">A10</font></i></a>, <a class="txt" href="funct_2.html#E3:12_1_1_3_2"><i><font color="Green" title="E9">A11</font></i></a>, <a class="txt" href="funct_2.html#E4:12_1_1_3_2"><i><font color="Green" title="E10">A12</font></i></a></i>;<br/>
<a NAME="E6:12_1_1_3_2"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">ff</font> where <font color="Olive" title="b1">ff</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c9">y</font>,<font color="Olive" title="b1">ff</font>] </span>}</span>  <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="funct_2.html#E3:12_1_1_3_2"><i><font color="Green" title="E9">A11</font></i></a></i>;<br/>
<a NAME="E7:12_1_1_3_2"/><b>then </b>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">ff</font> where <font color="Olive" title="b1">ff</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c9">y</font>,<font color="Olive" title="b1">ff</font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="funct_2.html#E9:12_1_1_3"><i><font color="Green" title="E6">A8</font></i></a>, <a class="txt" href="funct_2.html#E5:12_1_1_3_2"><i><font color="Green" title="E11">A13</font></i></a></i>;<br/>
<a NAME="E8:12_1_1_3_2"/><b>then </b>
 ex <font color="Olive" title="b1">ff</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> st <br/>( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">ff</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c9">y</font>,<font color="Olive" title="b1">ff</font>] )
 
;<br/>
<b>hence </b><a NAME="E9:12_1_1_3_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">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>then reconsider </b><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c7">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() <b>by </b><i><a class="txt" href="funct_2.html#E11:12_1_1_3"><i><font color="Green" title="E7">A9</font></i></a>, <a class="ref" href="funct_2.html#D1" target="_self" title="FUNCT_2:def.1">Def1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/><b>take </b>
<font color="Maroon" title="c8">f</font>
; <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> 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">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]</span><br/><b>let </b><font color="Maroon" title="c9">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="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>] )</span><br/><b>set </b><font color="Maroon" title="c10">x0</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">ff</font> where <font color="Olive" title="b1">ff</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">ff</font>] </span>}</span> ;<br/><b>assume </b><a NAME="E15:12_1_1_3"/><i><font color="Green" title="E9">A14</font></i>: 
<font color="Maroon" title="c9">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"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>]</span><br/><a NAME="E16:12_1_1_3"/><b>then </b><i><font color="Green" title="E10">A15</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">ff</font> where <font color="Olive" title="b1">ff</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">ff</font>] </span>}</span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">M</font>
 ;<br/><a NAME="E17:12_1_1_3"/>
<font color="Maroon" title="c8">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="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">ff</font> where <font color="Olive" title="b1">ff</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">ff</font>] </span>}</span> 
 <b>by </b><i><a class="txt" href="funct_2.html#E12:12_1_1_3"><i><font color="Green" title="E8">A10</font></i></a>, <a class="txt" href="funct_2.html#E15:12_1_1_3"><i><font color="Green" title="E9">A14</font></i></a></i>;<br/><a NAME="E18:12_1_1_3"/><b>then </b>
<font color="Maroon" title="c8">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#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">ff</font> where <font color="Olive" title="b1">ff</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">ff</font>] </span>}</span> 
 <b>by </b><i><a class="txt" href="funct_2.html#E9:12_1_1_3"><i><font color="Green" title="E6">A8</font></i></a>, <a class="txt" href="funct_2.html#E16:12_1_1_3"><i><font color="Green" title="E10">A15</font></i></a></i>;<br/><a NAME="E19:12_1_1_3"/><b>then </b>
 ex <font color="Olive" title="b1">ff</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> st <br/>( <font color="Olive" title="b1">ff</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">ff</font>] )
 ;<br/><b>hence </b><a NAME="E20:12_1_1_3"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</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></div><b>end;</b></div>

</div>
