<?xml version="1.0"?>
<div class="add">

<span class="kw">set </span><font color="Maroon" title="c1">X</font> = <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>6</sub></font>(<font color="Olive" title="b1">i</font>) where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Olive" title="b1">i</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">F<sub>4</sub></font>() </span> } </span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c2">Y</font> = <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>6</sub></font>(<font color="Olive" title="b1">j</font>) where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Olive" title="b1">j</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> <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="funct_7.html#K1" title="FUNCT_7:func.1">+*</a> (<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>())</span>)</span> </span> } </span> ;<br/>
<span class="kw">thus </span><a NAME="E1:182_1"/>
<span class="p1"> { <span class="default"> <font color="Maroon">F<sub>6</sub></font>(<font color="Olive" title="b1">i</font>) where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Olive" title="b1">i</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">F<sub>4</sub></font>() </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>6</sub></font>(<font color="Olive" title="b1">j</font>) where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Olive" title="b1">j</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> <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="funct_7.html#K1" title="FUNCT_7:func.1">+*</a> (<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>())</span>)</span> </span> } </span> 
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</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"> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>6</sub></font>(<font color="Olive" title="b1">j</font>) where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Olive" title="b1">j</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> <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="funct_7.html#K1" title="FUNCT_7:func.1">+*</a> (<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>())</span>)</span> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>6</sub></font>(<font color="Olive" title="b1">i</font>) where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Olive" title="b1">i</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">F<sub>4</sub></font>() </span> } </span> </span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="182_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">e</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="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>6</sub></font>(<font color="Olive" title="b1">i</font>) where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Olive" title="b1">i</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">F<sub>4</sub></font>() </span> } </span>  or <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>6</sub></font>(<font color="Olive" title="b1">j</font>) where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Olive" title="b1">j</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> <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="funct_7.html#K1" title="FUNCT_7:func.1">+*</a> (<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>())</span>)</span> </span> } </span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:182_1_1"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>6</sub></font>(<font color="Olive" title="b1">i</font>) where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Olive" title="b1">i</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">F<sub>4</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="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>6</sub></font>(<font color="Olive" title="b1">j</font>) where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Olive" title="b1">j</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> <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="funct_7.html#K1" title="FUNCT_7:func.1">+*</a> (<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>())</span>)</span> </span> } </span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<span class="kw"> such that </span><br/><a NAME="E3:182_1_1"/><span class="lab"><font color="Green" title="E84">A3</font></span>: 
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c4">j</font>)
 <span class="kw">and </span><br/><a NAME="E4:182_1_1"/><span class="lab"><font color="Green" title="E85">A4</font></span>: 
<font color="Maroon" title="c4">j</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">F<sub>4</sub></font>()
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:182_1_1"/><span class="lab"><font color="Green" title="E86">A5</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon">F<sub>4</sub></font>()
 <span class="kw">and </span><br/><a NAME="E7:182_1_1"/><span class="lab"><font color="Green" title="E87">A6</font></span>: 
<font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:182_1_1"><span class="lab"><font color="Green" title="E85">A4</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="E8:182_1_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="funct_7.html#K1" title="FUNCT_7:func.1">+*</a> (<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>())</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:182_1_1"><span class="lab"><font color="Green" title="E86">A5</font></span></a>, <a class="ref" href="funct_7.html#T30" target="_self" onmouseover="rs('funct_7/T30')" onmouseout="rh()">Th29</a></span>;<br/>
<a NAME="E9:182_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A7</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="funct_7.html#K1" title="FUNCT_7:func.1">+*</a> (<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>())</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</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> <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="funct_7.html#K1" title="FUNCT_7:func.1">+*</a> (<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>())</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="182_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="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="funct_7.html#K1" title="FUNCT_7:func.1">+*</a> (<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>())</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font></span>)</span>)</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="182_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="E1:182_1_1_1_1"/>
( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>() or <font color="Maroon" title="c5">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">F<sub>5</sub></font>() )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="182_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:182_1_1_1_1_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</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="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="funct_7.html#K1" title="FUNCT_7:func.1">+*</a> (<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>())</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font></span>)</span>)</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:182_1_1_1_1_1"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="funct_7.html#K1" title="FUNCT_7:func.1">+*</a> (<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>())</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:182"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E2:182"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E3:182_1_1"><span class="lab"><font color="Green" title="E84">A3</font></span></a>, <a class="txt" href="#E7:182_1_1"><span class="lab"><font color="Green" title="E87">A6</font></span></a>, <a class="ref" href="funct_7.html#T31" target="_self" onmouseover="rs('funct_7/T31')" onmouseout="rh()">Th30</a>, <a class="txt" href="#E6:182_1_1"><span class="lab"><font color="Green" title="E86">A5</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="182_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:182_1_1_1_1_2"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">F<sub>5</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="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="funct_7.html#K1" title="FUNCT_7:func.1">+*</a> (<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>())</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font></span>)</span>)</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:182_1_1_1_1_2"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="funct_7.html#K1" title="FUNCT_7:func.1">+*</a> (<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>())</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:182_1_1"><span class="lab"><font color="Green" title="E84">A3</font></span></a>, <a class="ref" href="funct_7.html#T32" target="_self" onmouseover="rs('funct_7/T32')" onmouseout="rh()">Th31</a>, <a class="txt" href="#E7:182_1_1"><span class="lab"><font color="Green" title="E87">A6</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></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="E11:182_1_1"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>6</sub></font>(<font color="Olive" title="b1">j</font>) where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Olive" title="b1">j</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> <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="funct_7.html#K1" title="FUNCT_7:func.1">+*</a> (<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>())</span>)</span> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:182_1_1"><span class="lab"><font color="Green" title="E88">A7</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">let </span><font color="Maroon" title="c3">e</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="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>6</sub></font>(<font color="Olive" title="b1">j</font>) where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Olive" title="b1">j</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> <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="funct_7.html#K1" title="FUNCT_7:func.1">+*</a> (<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>())</span>)</span> </span> } </span>  or <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>6</sub></font>(<font color="Olive" title="b1">i</font>) where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Olive" title="b1">i</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">F<sub>4</sub></font>() </span> } </span>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:182_1"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>6</sub></font>(<font color="Olive" title="b1">j</font>) where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Olive" title="b1">j</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> <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="funct_7.html#K1" title="FUNCT_7:func.1">+*</a> (<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>())</span>)</span> </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="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>6</sub></font>(<font color="Olive" title="b1">i</font>) where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Olive" title="b1">i</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">F<sub>4</sub></font>() </span> } </span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<span class="kw"> such that </span><br/><a NAME="E4:182_1"/><span class="lab"><font color="Green" title="E84">A8</font></span>: 
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c4">j</font>)
 <span class="kw">and </span><br/><a NAME="E5:182_1"/><span class="lab"><font color="Green" title="E85">A9</font></span>: 
<font color="Maroon" title="c4">j</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> <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="funct_7.html#K1" title="FUNCT_7:func.1">+*</a> (<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>())</span>)</span>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E7:182_1"/><span class="lab"><font color="Green" title="E86">A10</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="funct_7.html#K1" title="FUNCT_7:func.1">+*</a> (<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>())</span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:182_1"/><span class="lab"><font color="Green" title="E87">A11</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="funct_7.html#K1" title="FUNCT_7:func.1">+*</a> (<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>())</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:182_1"><span class="lab"><font color="Green" title="E85">A9</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="E9:182_1"/><span class="lab"><font color="Green" title="E88">A12</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon">F<sub>4</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:182_1"><span class="lab"><font color="Green" title="E86">A10</font></span></a>, <a class="ref" href="funct_7.html#T30" target="_self" onmouseover="rs('funct_7/T30')" onmouseout="rh()">Th29</a></span>;<br/>
<a NAME="E10:182_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A13</font></span>: 
<font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</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">F<sub>4</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="182_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"> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font></span>)</span>)</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="182_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="E1:182_1_2_1"/>
( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>() or <font color="Maroon" title="c5">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">F<sub>5</sub></font>() )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="182_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:182_1_2_1_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</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="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font></span>)</span>)</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:182_1_2_1_1"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:182"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E2:182"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E4:182_1"><span class="lab"><font color="Green" title="E84">A8</font></span></a>, <a class="txt" href="#E8:182_1"><span class="lab"><font color="Green" title="E87">A11</font></span></a>, <a class="ref" href="funct_7.html#T31" target="_self" onmouseover="rs('funct_7/T31')" onmouseout="rh()">Th30</a>, <a class="txt" href="#E9:182_1"><span class="lab"><font color="Green" title="E88">A12</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="182_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:182_1_2_1_2"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">F<sub>5</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="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font></span>)</span>)</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:182_1_2_1_2"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:182_1"><span class="lab"><font color="Green" title="E84">A8</font></span></a>, <a class="ref" href="funct_7.html#T32" target="_self" onmouseover="rs('funct_7/T32')" onmouseout="rh()">Th31</a>, <a class="txt" href="#E8:182_1"><span class="lab"><font color="Green" title="E87">A11</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="E12:182_1"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>6</sub></font>(<font color="Olive" title="b1">i</font>) where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Olive" title="b1">i</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">F<sub>4</sub></font>() </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:182_1"><span class="lab"><font color="Green" title="E89">A13</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>
