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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> for <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>()<br/>  for <font color="Olive" title="b2">h</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 holds <br/><font color="Olive" title="b2">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c1">F2</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c2">F1</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span> ;<br/>
<a NAME="E1:41_1"/><span class="lab"><font color="Green" title="E27">A5</font></span>: 
 for <font color="Olive" title="b1">f9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">f9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">f</font> where <font color="Olive" title="b2">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span>  holds <br/> ex <font color="Olive" title="b2">g9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">g9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b3">f</font> where <font color="Olive" title="b3">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b3">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b4">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b4">x</font>) ) ) </span> } </span>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">f9</font>,<font color="Olive" title="b2">g9</font>] )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="41_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">f9</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <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" title="c3">f9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span>  implies  ex <font color="Olive" title="b1">g9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">g9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">f</font> where <font color="Olive" title="b2">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b3">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b3">x</font>) ) ) </span> } </span>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">f9</font>,<font color="Olive" title="b1">g9</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:41_1_1"/>
<font color="Maroon" title="c3">f9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </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">  ex <font color="Olive" title="b1">g9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">g9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">f</font> where <font color="Olive" title="b2">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b3">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b3">x</font>) ) ) </span> } </span>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">f9</font>,<font color="Olive" title="b1">g9</font>] )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">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>()<span class="kw"> such that </span><br/><a NAME="E3:41_1_1"/><span class="lab"><font color="Green" title="E28">A6</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f9</font>
 <span class="kw">and </span><br/><a NAME="E4:41_1_1"/><span class="lab"><font color="Green" title="E29">A7</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()]
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">h</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()<span class="kw"> such that </span><br/><a NAME="E6:41_1_1"/><span class="lab"><font color="Green" title="E30">A8</font></span>: 
( <font color="Maroon" title="c5">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> &amp; (  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon" title="c5">h</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">F<sub>5</sub></font>(<font color="Olive" title="b1">x</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="stirl2_1.html#S2" target="_self" onmouseover="rs('stirl2_1/S2')" onmouseout="rh()">STIRL2_1:sch 2</a>(<a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E3:41"><span class="lab"><font color="Green" title="E25">A3</font></span></a>);<br/></span>
<a NAME="E7:41_1_1"/><span class="lab"><font color="Green" title="E31">A9</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">f9</font>,<font color="Maroon" title="c5">h</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_1_1"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="txt" href="#E6:41_1_1"><span class="lab"><font color="Green" title="E30">A8</font></span></a></span>;<br/>
<a NAME="E8:41_1_1"/><span class="lab"><font color="Green" title="E32">A10</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>()
 
;<br/>
<a NAME="E9:41_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">h</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="txt" href="#E4:41_1_1"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="txt" href="#E6:41_1_1"><span class="lab"><font color="Green" title="E30">A8</font></span></a></span>;<br/>
<a NAME="E10:41_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">h</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:41_1_1"><span class="lab"><font color="Green" title="E30">A8</font></span></a>, <a class="txt" href="#E8:41_1_1"><span class="lab"><font color="Green" title="E32">A10</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:41_1_1"/>
 ex <font color="Olive" title="b1">g9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">g9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">f</font> where <font color="Olive" title="b2">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b3">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b3">x</font>) ) ) </span> } </span>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">f9</font>,<font color="Olive" title="b1">g9</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:41_1_1"><span class="lab"><font color="Green" title="E31">A9</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/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c3">ff</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span> ,<span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </span> <span class="kw"> such that </span><br/><a NAME="E3:41_1"/><span class="lab"><font color="Green" title="E28">A11</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">f</font> where <font color="Olive" title="b2">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S1" onmouseover="rs('funct_2/S1')" onmouseout="rh()">FUNCT_2:sch 1</a>(<a class="txt" href="#E1:41_1"><span class="lab"><font color="Green" title="E27">A5</font></span></a>);<br/></span>
<a NAME="E4:41_1"/><span class="lab"><font color="Green" title="E29">A12</font></span>: 
( <font color="Maroon">F<sub>4</sub></font>() is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies <font color="Maroon">F<sub>3</sub></font>() is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E3:41"><span class="lab"><font color="Green" title="E25">A3</font></span></a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="41_1_2"><span class="kw">now </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">  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </span> </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:41_1_2_1"/>
( <font color="Maroon">F<sub>4</sub></font>() is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  not <font color="Maroon">F<sub>4</sub></font>() is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:41_1_2_1_1"/><span class="lab"><font color="Green" title="E30">A13</font></span>: 
<font color="Maroon">F<sub>4</sub></font>() is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <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="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </span> </span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c4">Empty</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/><a NAME="E2:41_1_2_1_1"/><span class="lab"><font color="Green" title="E31">A14</font></span>: 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="41_1_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <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"> (  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </span>  or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:41_1_2_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </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"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span><br/>

<a NAME="E2:41_1_2_1_1_1"/><span class="kw">then </span>
 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>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) )
 
;<br/>
<a NAME="E3:41_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_2_1_1"><span class="lab"><font color="Green" title="E30">A13</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:41_1_2_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</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/>


</div><span class="kw">end;</span></div><a NAME="E3:41_1_2_1_1"/><span class="lab"><font color="Green" title="E32">A15</font></span>: 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="41_1_2_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <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"> (  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span>  or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:41_1_2_1_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </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"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span><br/>

<a NAME="E2:41_1_2_1_1_2"/><span class="kw">then </span>
 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/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] )
 
;<br/>
<a NAME="E3:41_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E1:41_1_2_1_1"><span class="lab"><font color="Green" title="E30">A13</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:41_1_2_1_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</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/>


</div><span class="kw">end;</span></div><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="41_1_2_1_1_3"><span class="kw">now </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">  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </span> </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2_1_1_3_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:41_1_2_1_1_3_1"/>
( <font color="Maroon">P<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> , <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> , <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ] or  not <font color="Maroon">P<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> , <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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="41_1_2_1_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:41_1_2_1_1_3_1_1"/><span class="lab"><font color="Green" title="E33">A16</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> , <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> , <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 ; <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="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </span> </span><br/><div class="add"><a NAME="E2:41_1_2_1_1_3_1_1"/><span class="lab"><font color="Green" title="E34">A17</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E3:41_1_2_1_1_3_1_1"/><span class="lab"><font color="Green" title="E35">A18</font></span>: 
<font color="Maroon">F<sub>2</sub></font>() is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E1:41_1_2_1_1"><span class="lab"><font color="Green" title="E30">A13</font></span></a></span>;<br/><a NAME="E4:41_1_2_1_1_3_1_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is   <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>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E3:41_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E35">A18</font></span></a>, <a class="txt" href="#E2:41_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E34">A17</font></span></a>, <a class="ref" href="funct_2.html#T1" onmouseover="rs('funct_2/T1')" onmouseout="rh()">FUNCT_2:1</a></span>;<br/><a NAME="E5:41_1_2_1_1_3_1_1"/><span class="kw">then </span>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E1:41_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="txt" href="#E3:41_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E35">A18</font></span></a></span>;<br/><a NAME="E6:41_1_2_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A19</font></span>: 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_1_2_1_1"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" onmouseover="rs('zfmisc_1/T33')" onmouseout="rh()">ZFMISC_1:33</a></span>;<br/><a NAME="E7:41_1_2_1_1_3_1_1"/><span class="lab"><font color="Green" title="E37">A20</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">Empty</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_1"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="txt" href="#E1:41_1_2_1_1"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="txt" href="#E7:41_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E37">A20</font></span></a>, <a class="ref" href="funct_2.html#T1" onmouseover="rs('funct_2/T1')" onmouseout="rh()">FUNCT_2:1</a></span>;<br/><a NAME="E9:41_1_2_1_1_3_1_1"/><span class="lab"><font color="Green" title="E38">A21</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon" title="c5">Empty</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">F<sub>5</sub></font>(<font color="Olive" title="b1">x</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_1"><span class="lab"><font color="Green" title="E29">A12</font></span></a></span>;<br/><a NAME="E10:41_1_2_1_1_3_1_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Empty</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>()
 ;<br/><a NAME="E11:41_1_2_1_1_3_1_1"/><span class="kw">then </span>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_1"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="txt" href="#E1:41_1_2_1_1"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="txt" href="#E1:41_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="txt" href="#E9:41_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E38">A21</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E12:41_1_2_1_1_3_1_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_1_2_1_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="txt" href="#E6:41_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" onmouseover="rs('zfmisc_1/T33')" onmouseout="rh()">ZFMISC_1:33</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2_1_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:41_1_2_1_1_3_1_2"/><span class="lab"><font color="Green" title="E33">A22</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> , <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> , <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 ; <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="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </span> </span><br/><div class="add"><a NAME="E2:41_1_2_1_1_3_1_2"/><span class="lab"><font color="Green" title="E34">A23</font></span>: 
 not  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="41_1_2_1_1_3_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:41_1_2_1_1_3_1_2_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </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"> contradiction</span><br/>

<a NAME="E2:41_1_2_1_1_3_1_2_1"/><span class="kw">then </span>
 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/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:41_1_2_1_1_3_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E3:41"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E1:41_1_2_1_1"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="txt" href="#E1:41_1_2_1_1_3_1_2"><span class="lab"><font color="Green" title="E33">A22</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/>


</div><span class="kw">end;</span></div><a NAME="E3:41_1_2_1_1_3_1_2"/><span class="lab"><font color="Green" title="E35">A24</font></span>: 
( <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_1_2_1_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" onmouseover="rs('zfmisc_1/T33')" onmouseout="rh()">ZFMISC_1:33</a></span>;<br/><a NAME="E4:41_1_2_1_1_3_1_2"/><span class="lab"><font color="Green" title="E36">A25</font></span>: 
 not  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </span> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="41_1_2_1_1_3_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:41_1_2_1_1_3_1_2_2"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </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"> contradiction</span><br/>

<a NAME="E2:41_1_2_1_1_3_1_2_2"/><span class="kw">then </span>
 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>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:41_1_2_1_1_3_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_1"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="txt" href="#E1:41_1_2_1_1"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="txt" href="#E1:41_1_2_1_1_3_1_2"><span class="lab"><font color="Green" title="E33">A22</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/>


</div><span class="kw">end;</span></div><a NAME="E5:41_1_2_1_1_3_1_2"/>
( <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_1_2_1_1"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" onmouseover="rs('zfmisc_1/T33')" onmouseout="rh()">ZFMISC_1:33</a></span>;<br/><span class="kw">hence </span><a NAME="E6:41_1_2_1_1_3_1_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_1_2_1_1_3_1_2"><span class="lab"><font color="Green" title="E34">A23</font></span></a>, <a class="txt" href="#E4:41_1_2_1_1_3_1_2"><span class="lab"><font color="Green" title="E36">A25</font></span></a>, <a class="txt" href="#E3:41_1_2_1_1_3_1_2"><span class="lab"><font color="Green" title="E35">A24</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:41_1_2_1_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:41_1_2_1_2"/><span class="lab"><font color="Green" title="E30">A26</font></span>: 
 not <font color="Maroon">F<sub>4</sub></font>() is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <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="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </span> </span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="41_1_2_1_2_1"><span class="kw">now </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">  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </span> </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:41_1_2_1_2_1_1"/>
( <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </span>  is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  not <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </span>  is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:41_1_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E31">A27</font></span>: 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </span>  is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <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="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </span> </span><br/><div class="add"><a NAME="E2:41_1_2_1_2_1_1_1"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span>  is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="41_1_2_1_2_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:41_1_2_1_2_1_1_1_1"/>
 not <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span>  is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <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"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:41_1_2_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E32">A28</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span> 
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">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>()<span class="kw"> such that </span><br/><a NAME="E5:41_1_2_1_2_1_1_1_1"/>
<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">and </span><br/><a NAME="E6:41_1_2_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E33">A29</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_1_2_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E32">A28</font></span></a></span>;<br/>
<a NAME="E7:41_1_2_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E34">A30</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>()
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">h</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()<span class="kw"> such that </span><br/><a NAME="E9:41_1_2_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E35">A31</font></span>: 
( <font color="Maroon" title="c6">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> &amp; (  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon" title="c6">h</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">F<sub>5</sub></font>(<font color="Olive" title="b1">x</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="stirl2_1.html#S2" target="_self" onmouseover="rs('stirl2_1/S2')" onmouseout="rh()">STIRL2_1:sch 2</a>(<a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E3:41"><span class="lab"><font color="Green" title="E25">A3</font></span></a>);<br/></span>
<a NAME="E10:41_1_2_1_2_1_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">h</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="txt" href="#E6:41_1_2_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E33">A29</font></span></a>, <a class="txt" href="#E9:41_1_2_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E35">A31</font></span></a></span>;<br/>
<a NAME="E11:41_1_2_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">h</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:41_1_2_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E35">A31</font></span></a>, <a class="txt" href="#E7:41_1_2_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E34">A30</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:41_1_2_1_2_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E31">A27</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/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:41_1_2_1_2_1_1_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E31">A27</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:41_1_2_1_2_1_1_2"/><span class="lab"><font color="Green" title="E31">A32</font></span>: 
 not <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </span>  is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <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="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </span> </span><br/><div class="add"><a NAME="E2:41_1_2_1_2_1_1_2"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">ff</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="41_1_2_1_2_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <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"> (  not <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </span>  or <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">ff</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:41_1_2_1_2_1_1_2_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </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"> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">ff</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()<span class="kw"> such that </span><br/><a NAME="E3:41_1_2_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E32">A33</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font>
 <span class="kw">and </span><br/><a NAME="E4:41_1_2_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E33">A34</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()]
 <span class="kw">and </span><br/><a NAME="E5:41_1_2_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E34">A35</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>()
 <span class="kw">and </span><br/><a NAME="E6:41_1_2_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E35">A36</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon" title="c5">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">F<sub>5</sub></font>(<font color="Olive" title="b1">x</font>)
 ;<br/>
<a NAME="E7:41_1_2_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E36">A37</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon">F<sub>1</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>;<br/>
<a NAME="E8:41_1_2_1_2_1_1_2_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_2_1_2"><span class="lab"><font color="Green" title="E30">A26</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E9:41_1_2_1_2_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A38</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <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="#E2:41"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E7:41_1_2_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E36">A37</font></span></a>, <a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">h</font> = <font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>() as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T1" onmouseover="rs('funct_2/T1')" onmouseout="rh()">FUNCT_2:1</a></span>;<br/>
<a NAME="E11:41_1_2_1_2_1_1_2_1"/>
(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies <font color="Maroon">F<sub>1</sub></font>() is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:41_1_2_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E37">A38</font></span></a>, <a class="ref" href="relat_1.html#T42" onmouseover="rs('relat_1/T42')" onmouseout="rh()">RELAT_1:42</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">h</font> = <font color="Maroon" title="c6">h</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>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:41_1_2_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E34">A35</font></span></a>, <a class="ref" href="funct_2.html#T6" onmouseover="rs('funct_2/T6')" onmouseout="rh()">FUNCT_2:6</a></span>;<br/>
<a NAME="E13:41_1_2_1_2_1_1_2_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="txt" href="#E4:41_1_2_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E33">A34</font></span></a>, <a class="txt" href="#E6:41_1_2_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E35">A36</font></span></a></span>;<br/>
<a NAME="E14:41_1_2_1_2_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A39</font></span>: 
<font color="Maroon" title="c7">h</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span> 
 
;<br/>
<a NAME="E15:41_1_2_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E39">A40</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">ff</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E31">A32</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E16:41_1_2_1_2_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">h</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">ff</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:41_1_2_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E38">A39</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E17:41_1_2_1_2_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">h</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </span> 
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">ffh</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()<span class="kw"> such that </span><br/><a NAME="E19:41_1_2_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E40">A41</font></span>: 
<font color="Maroon" title="c8">ffh</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">h</font>
 <span class="kw">and </span><br/><a NAME="E20:41_1_2_1_2_1_1_2_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c8">ffh</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()]
 <span class="kw">and </span><br/><a NAME="E21:41_1_2_1_2_1_1_2_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">ffh</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>()
 <span class="kw">and </span><br/><a NAME="E22:41_1_2_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E41">A42</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon" title="c8">ffh</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">F<sub>5</sub></font>(<font color="Olive" title="b1">x</font>)
 ;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="41_1_2_1_2_1_1_2_1_1"><span class="kw">now </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">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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>3</sub></font>() holds <br/><font color="Maroon" title="c8">ffh</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" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <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" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() implies <font color="Maroon" title="c8">ffh</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="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:41_1_2_1_2_1_1_2_1_1"/><span class="lab"><font color="Green" title="E42">A43</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</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" title="c8">ffh</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="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="41_1_2_1_2_1_1_2_1_1_1"><span class="kw">now </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"> <font color="Maroon" title="c8">ffh</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="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2_1_2_1_1_2_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:41_1_2_1_2_1_1_2_1_1_1_1"/>
( <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>() or  not <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>() )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2_1_2_1_1_2_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:41_1_2_1_2_1_1_2_1_1_1_1_1"/>
<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" 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" title="c8">ffh</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="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span><br/><div class="add"><a NAME="E2:41_1_2_1_2_1_1_2_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A44</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E3:41_1_2_1_2_1_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c8">ffh</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_1"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="txt" href="#E14:41_1_2_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E38">A39</font></span></a>, <a class="txt" href="#E19:41_1_2_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E40">A41</font></span></a></span>;<br/><a NAME="E4:41_1_2_1_2_1_1_2_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">h</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="c8">ffh</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_1_2_1_2_1_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A44</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a></span>;<br/><span class="kw">hence </span><a NAME="E5:41_1_2_1_2_1_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c8">ffh</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="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_1_2_1_2_1_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A44</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2_1_2_1_1_2_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:41_1_2_1_2_1_1_2_1_1_1_1_2"/>
 not <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" 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" title="c8">ffh</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="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span><br/><div class="add"><a NAME="E2:41_1_2_1_2_1_1_2_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A45</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_2_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E42">A43</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E3:41_1_2_1_2_1_1_2_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">ffh</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">F<sub>5</sub></font>(<font color="Maroon" title="c9">x</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:41_1_2_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E41">A42</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:41_1_2_1_2_1_1_2_1_1_1_1_2"/>
<font color="Maroon" title="c8">ffh</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="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:41_1_2_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E35">A36</font></span></a>, <a class="txt" href="#E2:41_1_2_1_2_1_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E43">A45</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:41_1_2_1_2_1_1_2_1_1"/>
<font color="Maroon" title="c8">ffh</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="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</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"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E24:41_1_2_1_2_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">ffh</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T12" onmouseover="rs('funct_2/T12')" onmouseout="rh()">FUNCT_2:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E25:41_1_2_1_2_1_1_2_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">ff</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_1_2_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E32">A33</font></span></a>, <a class="txt" href="#E14:41_1_2_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E38">A39</font></span></a>, <a class="txt" href="#E15:41_1_2_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E39">A40</font></span></a>, <a class="txt" href="#E19:41_1_2_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E40">A41</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</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/>


</div><span class="kw">end;</span></div><a NAME="E3:41_1_2_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A46</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">ff</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </span> 
 ;<br/><a NAME="E4:41_1_2_1_2_1_1_2"/>
 for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b3">f</font> where <font color="Olive" title="b3">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b3">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span>  &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b3">f</font> where <font color="Olive" title="b3">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b3">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span>  &amp; <font color="Maroon" title="c3">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="41_1_2_1_2_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c5">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <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" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span>  &amp; <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span>  &amp; <font color="Maroon" title="c3">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x2</font> implies <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:41_1_2_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E33">A47</font></span>: 
<font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span> 
 <span class="kw">and </span><br/><a NAME="E2:41_1_2_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E34">A48</font></span>: 
<font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span> 
 <span class="kw">and </span><br/><a NAME="E3:41_1_2_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E35">A49</font></span>: 
<font color="Maroon" title="c3">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x2</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" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x2</font></span><br/>

<a NAME="E4:41_1_2_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E36">A50</font></span>: 
 ex <font color="Olive" title="b1">f2</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/>( <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f2</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f2</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_1_2_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E34">A48</font></span></a></span>;<br/>
<a NAME="E5:41_1_2_1_2_1_1_2_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">ff</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E31">A32</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E6:41_1_2_1_2_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">ff</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_2_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E33">A47</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E7:41_1_2_1_2_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </span> 
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">F1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()<span class="kw"> such that </span><br/><a NAME="E9:41_1_2_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E37">A51</font></span>: 
<font color="Maroon" title="c3">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F1</font>
 <span class="kw">and </span><br/><a NAME="E10:41_1_2_1_2_1_1_2_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">F1</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()]
 <span class="kw">and </span><br/><a NAME="E11:41_1_2_1_2_1_1_2_2"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>()
 <span class="kw">and </span><br/><a NAME="E12:41_1_2_1_2_1_1_2_2"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon" title="c6">F1</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">F<sub>5</sub></font>(<font color="Olive" title="b1">x</font>)
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">f1</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>()<span class="kw"> such that </span><br/><a NAME="E14:41_1_2_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E38">A52</font></span>: 
<font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f1</font>
 <span class="kw">and </span><br/><a NAME="E15:41_1_2_1_2_1_1_2_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c7">f1</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_2_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E33">A47</font></span></a></span>;<br/>
<a NAME="E16:41_1_2_1_2_1_1_2_2"/>
<font color="Maroon" title="c6">F1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_1"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="txt" href="#E1:41_1_2_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E33">A47</font></span></a>, <a class="txt" href="#E14:41_1_2_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E38">A52</font></span></a>, <a class="txt" href="#E9:41_1_2_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E37">A51</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:41_1_2_1_2_1_1_2_2"/>
<font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_1"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="txt" href="#E2:41_1_2_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E34">A48</font></span></a>, <a class="txt" href="#E3:41_1_2_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E35">A49</font></span></a>, <a class="txt" href="#E14:41_1_2_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E38">A52</font></span></a>, <a class="txt" href="#E4:41_1_2_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E36">A50</font></span></a>, <a class="txt" href="#E9:41_1_2_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E37">A51</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/>


</div><span class="kw">end;</span></div><a NAME="E5:41_1_2_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A53</font></span>: 
<font color="Maroon" title="c3">ff</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E31">A32</font></span></a>, <a class="ref" href="funct_2.html#T19" onmouseover="rs('funct_2/T19')" onmouseout="rh()">FUNCT_2:19</a></span>;<br/><a NAME="E6:41_1_2_1_2_1_1_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">ff</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E31">A32</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E7:41_1_2_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span> ,<span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </span>  <a href="tarski.html#R3" title="TARSKI:pred.3">are_equipotent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_1_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E32">A46</font></span></a>, <a class="txt" href="#E5:41_1_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E33">A53</font></span></a>, <a class="ref" href="wellord2.html#D4" onmouseover="rs('wellord2/D4')" onmouseout="rh()">WELLORD2:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E8:41_1_2_1_2_1_1_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T5" onmouseover="rs('card_1/T5')" onmouseout="rh()">CARD_1:5</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:41_1_2_1_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:41_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <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>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span> } </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/>


</div>
