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<b>consider </b><font color="Maroon">c<sub>1</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E2:93_1"/><i><font color="Green">E28</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>and </b><br/><a NAME="E3:93_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/>( ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) &amp; ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="partfun1.html#S1">PARTFUN1:sch 1</a>();<br/></i>
<div><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:93_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>assume </b><a NAME="E2:93_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>consider </b><font color="Maroon">c<sub>2</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><a NAME="E4:93_1_1"/>
( ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>] implies <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>2</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>] implies <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>2</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>() ) )
 <b>by </b><i><a class="txt" href="funct_2.html#E1:93"><i><font color="Green">E27</font></i></a>, <a class="txt" href="funct_2.html#E2:93_1_1"><i><font color="Green">E32</font></i></a></i>;<br/><b>hence </b><a NAME="E5:93_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="funct_2.html#E1:93_1_1"><i><font color="Green">E31</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:93_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>2</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:93_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:93_1_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E4:93_1_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:93_1_2"/>
( ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>] implies <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>3</sub></font>) ) &amp; ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>] implies <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>3</sub></font>) ) )
 
<b>by </b><i><a class="txt" href="funct_2.html#E2:93_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="funct_2.html#E3:93_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="funct_2.html#E3:93_1_2"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E6:93_1_2"/><b>then </b>
( ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>] implies <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>] implies <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>() ) )
 
<b>by </b><i><a class="txt" href="funct_2.html#E1:93"><i><font color="Green">E27</font></i></a>, <a class="txt" href="funct_2.html#E2:93_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="funct_2.html#E3:93_1_2"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:93_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i><a class="txt" href="funct_2.html#E4:93_1_2"><i><font color="Green">E32</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:93_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> is   <a href="funct_2.html#NM1">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#E2:93_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="funct_2.html#E4:93_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_2.html#D1" target="_self">Def1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>hence </b><a NAME="E7:93_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">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">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/>( ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] implies <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>2</sub></font>) ) &amp; ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] implies <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>2</sub></font>) ) )
 <b>by </b><i><a class="txt" href="funct_2.html#E3:93_1"><i><font color="Green">E29</font></i></a></i>;<br/>


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