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<a NAME="E1:1_1"/><span class="lab"><font color="Green" title="E3">A3</font></span>: 
 for <font color="Olive" title="b1">c</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   holds <br/>(  not <font color="Olive" title="b1">c</font> <a href="tarski.html#R2" title="TARSKI: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>] )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1"><span class="lab"><font color="Green" title="E2">A2</font></span></a></span>;<br/>
<a NAME="E2:1_1"/><span class="lab"><font color="Green" title="E4">A4</font></span>: 
 for <font color="Olive" title="b1">c</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">c</font> <a href="tarski.html#R2" title="TARSKI: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>] ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E1">A1</font></span></a></span>;<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#NK1" title="RELAT_1:NK.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#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">c</font> <a href="tarski.html#R2" title="TARSKI: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>) ) ) ) )
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="recdef_2.html#S1" onmouseover="rs('recdef_2/S1')" onmouseout="rh()">RECDEF_2:sch 1</a>(<a class="txt" href="#E2:1_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a>, <a class="txt" href="#E1:1_1"><span class="lab"><font color="Green" title="E3">A3</font></span></a>);<br/></span>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c1">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E5:1_1"/><span class="lab"><font color="Green" title="E5">A5</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.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>()
 <span class="kw">and </span><br/><a NAME="E6:1_1"/><span class="lab"><font color="Green" title="E6">A6</font></span>: 
 for <font color="Olive" title="b1">c</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">c</font> <a href="tarski.html#R2" title="TARSKI: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/>
<span class="kw">take </span>
<font color="Maroon" title="c1">f</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="relat_1.html#NK1" title="RELAT_1:NK.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>() &amp; (  for <font color="Olive" title="b1">c</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 <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>) ) ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E7:1_1"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.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>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1"><span class="lab"><font color="Green" title="E5">A5</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">c</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 <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>) ) )</span><br/>

<span class="kw">let </span><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>(); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <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>) ) )</span><br/>

<span class="kw">thus </span><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>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1"><span class="lab"><font color="Green" title="E6">A6</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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