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<a NAME="E1:5_1"/><i><font color="Green" title="E7">A5</font></i>: 
for <font color="Olive" title="b1">x</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">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<b>by </b><i><a class="txt" href="transgeo.html#E1:5"><i><font color="Green" title="E3">A1</font></i></a></i>;<br/>
<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>1</sub></font>()<b> such that </b><br/><a NAME="E3:5_1"/><i><font color="Green" title="E8">A6</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <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_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="transgeo.html#E1:5_1"><i><font color="Green" title="E7">A5</font></i></a>);<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">x1</font>, <font color="Maroon" title="c3">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:5_1_1_1"/><i><font color="Green" title="E9">A7</font></i>: 
( <font color="Maroon" title="c2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() )
 ;<br/><a NAME="E2:5_1_1_1"/><b>then </b>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x1</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">x1</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x2</font>] &amp; <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() )
 <b>by </b><i><a class="txt" href="transgeo.html#E3:5_1"><i><font color="Green" title="E8">A6</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/><b>hence </b><a NAME="E3:5_1_1_1"/>
( <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x1</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="c3">x2</font> implies <font color="Maroon" title="c2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x2</font> )
 <b>by </b><i><a class="txt" href="transgeo.html#E3:5"><i><font color="Green" title="E5">A3</font></i></a>, <a class="txt" href="transgeo.html#E1:5_1_1_1"><i><font color="Green" title="E9">A7</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:5_1_1"/>
<font color="Maroon" title="c1">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_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_2"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:5_1_1_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:5_1_1_2_1"/><i><font color="Green" title="E9">A8</font></i>: 
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/><b>then consider </b><font color="Maroon" title="c3">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:5_1_1_2_1"/><i><font color="Green" title="E10">A9</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">y</font>]
 <b>by </b><i><a class="txt" href="transgeo.html#E2:5"><i><font color="Green" title="E4">A2</font></i></a></i>;<br/><a NAME="E4:5_1_1_2_1"/>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font>] &amp; <font color="Maroon" title="c1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() )
 <b>by </b><i><a class="txt" href="transgeo.html#E3:5_1"><i><font color="Green" title="E8">A6</font></i></a></i>;<br/><a NAME="E5:5_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> &amp; <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>  )
 <b>by </b><i><a class="txt" href="transgeo.html#E4:5"><i><font color="Green" title="E6">A4</font></i></a>, <a class="txt" href="transgeo.html#E1:5_1_1_2_1"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="transgeo.html#E3:5_1_1_2_1"><i><font color="Green" title="E10">A9</font></i></a></i>;<br/><b>hence </b><a NAME="E6:5_1_1_2_1"/>
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:5_1_1_2"/>
<font color="Maroon">F<sub>1</sub></font>() <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:5_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</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="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c2">f</font> = <font color="Maroon" title="c1">f</font> as   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="funct_2.html#T83" title="FUNCT_2:th.83">FUNCT_2:83</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c2">f</font>
;<br/>

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>thus </b><a NAME="E6:5_1"/>
( <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>] )
 <b>by </b><i><a class="txt" href="transgeo.html#E3:5_1"><i><font color="Green" title="E8">A6</font></i></a></i>;<br/>

<b>assume </b><a NAME="E7:5_1"/><i><font color="Green" title="E9">A10</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>]
 ;<br/>

<a NAME="E8:5_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font>]
 
<b>by </b><i><a class="txt" href="transgeo.html#E3:5_1"><i><font color="Green" title="E8">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:5_1"/>
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="txt" href="transgeo.html#E4:5"><i><font color="Green" title="E6">A4</font></i></a>, <a class="txt" href="transgeo.html#E7:5_1"><i><font color="Green" title="E9">A10</font></i></a></i>;<br/>


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