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<a NAME="E1:1_1"/><i><font color="Green" title="E3">A3</font></i>: 
for <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">c</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">c</font>] implies  not <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">c</font>] ) &amp; ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">c</font>] implies  not <font color="Maroon">P<sub>3</sub></font>[<font color="Olive" title="b1">c</font>] ) &amp; ( <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">c</font>] implies  not <font color="Maroon">P<sub>3</sub></font>[<font color="Olive" title="b1">c</font>] ) )
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:1"><i><font color="Green" title="E1">A1</font></i></a></i>;<br/>
<a NAME="E2:1_1"/><i><font color="Green" title="E4">A4</font></i>: 
for <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() or <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">c</font>] or <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">c</font>] or <font color="Maroon">P<sub>3</sub></font>[<font color="Olive" title="b1">c</font>] )
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E2:1"><i><font color="Green" title="E2">A2</font></i></a></i>;<br/>
<a NAME="E3:1_1"/>
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">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="b2">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">c</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">c</font>] implies <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b2">c</font>) ) &amp; ( <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b2">c</font>] implies <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b2">c</font>) ) &amp; ( <font color="Maroon">P<sub>3</sub></font>[<font color="Olive" title="b2">c</font>] implies <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b2">c</font>) ) ) ) )
 
<b>from </b><i><a class="ref" href="recdef_2.html#S1" title="RECDEF_2:sch.1">RECDEF_2:sch 1</a>(<a class="txt" href="borsuk_6.html#E1:1_1"><i><font color="Green" title="E3">A3</font></i></a>, <a class="txt" href="borsuk_6.html#E2:1_1"><i><font color="Green" title="E4">A4</font></i></a>);<br/></i>
<b>then 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="E5:1_1"/><i><font color="Green" title="E5">A5</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">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>and </b><br/><a NAME="E6:1_1"/><i><font color="Green" title="E6">A6</font></i>: 
for <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">c</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">c</font>] implies <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">c</font>) ) &amp; ( <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">c</font>] implies <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">c</font>) ) &amp; ( <font color="Maroon">P<sub>3</sub></font>[<font color="Olive" title="b1">c</font>] implies <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">c</font>) ) )
 ;<br/>
<b>take </b>
<font color="Maroon" title="c1">f</font>
;<br/>

<b>thus </b><a NAME="E7:1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">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="borsuk_6.html#E5:1_1"><i><font color="Green" title="E5">A5</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c2">c</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="E8:1_1"/>
( ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">c</font>] implies <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c2">c</font>) ) &amp; ( <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c2">c</font>] implies <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c2">c</font>) ) &amp; ( <font color="Maroon">P<sub>3</sub></font>[<font color="Maroon" title="c2">c</font>] implies <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c2">c</font>) ) )
 <b>by </b><i><a class="txt" href="borsuk_6.html#E6:1_1"><i><font color="Green" title="E6">A6</font></i></a></i>;<br/>


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