<?xml version="1.0"?>
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<span class="kw">let </span><font color="Maroon" title="c1">R</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>   <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>   <a href="dilworth.html#V4" title="DILWORTH:attr.4">with_finite_stability#</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="dilworth.html#NM2" title="DILWORTH:NM.2">StableSet</a> of <font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b2">U</font> being  <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="dilworth.html#K4" title="DILWORTH:func.4">Upper</a> <font color="Olive" title="b1">A</font></span>)</span></span>)</span><br/> for <font color="Olive" title="b3">L</font> being  <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="dilworth.html#K3" title="DILWORTH:func.3">Lower</a> <font color="Olive" title="b1">A</font></span>)</span></span>)</span>  st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c1">R</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b2">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c1">R</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c1">R</font> holds <br/> ex <font color="Olive" title="b4">C</font> being  <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c1">R</font> st  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c1">R</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="dilworth.html#NM2" title="DILWORTH:NM.2">StableSet</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">U</font> being  <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="dilworth.html#K4" title="DILWORTH:func.4">Upper</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span><br/> for <font color="Olive" title="b2">L</font> being  <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="dilworth.html#K3" title="DILWORTH:func.3">Lower</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>  st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c1">R</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c1">R</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c1">R</font> holds <br/> ex <font color="Olive" title="b3">C</font> being  <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c1">R</font> st  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c1">R</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">U</font> be   <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="dilworth.html#K4" title="DILWORTH:func.4">Upper</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">L</font> being  <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="dilworth.html#K3" title="DILWORTH:func.3">Lower</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>  st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c1">R</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c3">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c1">R</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c1">R</font> holds <br/> ex <font color="Olive" title="b2">C</font> being  <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c1">R</font> st  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c1">R</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">L</font> be   <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="dilworth.html#K3" title="DILWORTH:func.3">Lower</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c1">R</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c3">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c1">R</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c4">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c1">R</font> implies  ex <font color="Olive" title="b1">C</font> being  <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c1">R</font> st  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c1">R</font> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:90"/><span class="lab"><font color="Green" title="E52">A1</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">and </span><br/><a NAME="E2:90"/><span class="lab"><font color="Green" title="E53">B1</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c3">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">and </span><br/><a NAME="E3:90"/><span class="lab"><font color="Green" title="E54">C1</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c4">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c1">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">C</font> being  <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c1">R</font> st  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c1">R</font></span><br/>

<a NAME="E4:90"/><span class="lab"><font color="Green" title="E55">D1</font></span>: 
not <font color="Maroon" title="c2">A</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="#E1:90"><span class="lab"><font color="Green" title="E52">A1</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">aA</font> =  <a href="dilworth.html#K4" title="DILWORTH:func.4">Upper</a> <font color="Maroon" title="c2">A</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">bA</font> =  <a href="dilworth.html#K3" title="DILWORTH:func.3">Lower</a> <font color="Maroon" title="c2">A</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">cR</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">Pa</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1">(<span class="default"><a href="dilworth.html#K4" title="DILWORTH:func.4">Upper</a> <font color="Maroon" title="c2">A</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">Pb</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1">(<span class="default"><a href="dilworth.html#K3" title="DILWORTH:func.3">Lower</a> <font color="Maroon" title="c2">A</font></span>)</span>;<br/>
<a NAME="E5:90"/><span class="lab"><font color="Green" title="E56">S2d</font></span>: 
 <a href="dilworth.html#K4" title="DILWORTH:func.4">Upper</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="dilworth.html#K4" title="DILWORTH:func.4">Upper</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></span>;<br/>
<a NAME="E6:90"/><span class="lab"><font color="Green" title="E57">S2e</font></span>: 
 <a href="dilworth.html#K3" title="DILWORTH:func.3">Lower</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="dilworth.html#K3" title="DILWORTH:func.3">Lower</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">Pa</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1">(<span class="default"><a href="dilworth.html#K4" title="DILWORTH:func.4">Upper</a> <font color="Maroon" title="c2">A</font></span>)</span> as  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>   <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>   <a href="dilworth.html#V4" title="DILWORTH:attr.4">with_finite_stability#</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:90"><span class="lab"><font color="Green" title="E55">D1</font></span></a></span>;<br/>
<a NAME="E8:90"/>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="dilworth.html#K4" title="DILWORTH:func.4">Upper</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T21" title="XBOOLE_1:th.21">XBOOLE_1:21</a></span>;<br/>
<a NAME="E9:90"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">D1a</font></span>: 
<font color="Maroon" title="c2">A</font> is   <a href="dilworth.html#NM2" title="DILWORTH:NM.2">StableSet</a> of <font color="Maroon" title="c10">Pa</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#T31" target="_self" title="DILWORTH:th.31">SPAChain</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">U</font><span class="kw"> such that </span><br/><a NAME="E11:90"/><span class="lab"><font color="Green" title="E59">E1</font></span>: 
<font color="Maroon" title="c11">f</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 <span class="kw">and </span><br/><a NAME="E12:90"/><span class="lab"><font color="Green" title="E60">E1a</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E2:90"><span class="lab"><font color="Green" title="E53">B1</font></span></a>, <a class="txt" href="#E9:90"><span class="lab"><font color="Green" title="E58">D1a</font></span></a>, <a class="ref" href="dilworth.html#T47" target="_self" title="DILWORTH:th.47">ACpart1</a></span>;<br/>
<a NAME="E13:90"/><span class="lab"><font color="Green" title="E61">E1b</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:90"><span class="lab"><font color="Green" title="E55">D1</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E14:90"/><span class="lab"><font color="Green" title="E62">E1d</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:90"><span class="lab"><font color="Green" title="E59">E1</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">Pb</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1">(<span class="default"><a href="dilworth.html#K3" title="DILWORTH:func.3">Lower</a> <font color="Maroon" title="c2">A</font></span>)</span> as  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>   <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>   <a href="dilworth.html#V4" title="DILWORTH:attr.4">with_finite_stability#</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:90"><span class="lab"><font color="Green" title="E55">D1</font></span></a></span>;<br/>
<a NAME="E16:90"/>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="dilworth.html#K3" title="DILWORTH:func.3">Lower</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T21" title="XBOOLE_1:th.21">XBOOLE_1:21</a></span>;<br/>
<a NAME="E17:90"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">F1a</font></span>: 
<font color="Maroon" title="c2">A</font> is   <a href="dilworth.html#NM2" title="DILWORTH:NM.2">StableSet</a> of <font color="Maroon" title="c12">Pb</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#T31" target="_self" title="DILWORTH:th.31">SPAChain</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">L</font><span class="kw"> such that </span><br/><a NAME="E19:90"/><span class="lab"><font color="Green" title="E64">G1</font></span>: 
<font color="Maroon" title="c13">g</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 <span class="kw">and </span><br/><a NAME="E20:90"/><span class="lab"><font color="Green" title="E65">G1a</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E3:90"><span class="lab"><font color="Green" title="E54">C1</font></span></a>, <a class="txt" href="#E17:90"><span class="lab"><font color="Green" title="E63">F1a</font></span></a>, <a class="ref" href="dilworth.html#T47" target="_self" title="DILWORTH:th.47">ACpart1</a></span>;<br/>
<a NAME="E21:90"/><span class="lab"><font color="Green" title="E66">G1b</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:90"><span class="lab"><font color="Green" title="E55">D1</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E22:90"/><span class="lab"><font color="Green" title="E67">G1d</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:90"><span class="lab"><font color="Green" title="E64">G1</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">h</font> = <font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">C</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span>;<br/>
<a NAME="E23:90"/><span class="lab"><font color="Green" title="E68">I1a</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</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="c11">f</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lexbfs.html#D2" title="LEXBFS:def.2">LEXBFS:def 2</a></span>;<br/>
<a NAME="E24:90"/><span class="lab"><font color="Green" title="E69">Y1a</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span> or <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:90_1"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:90_1"/><span class="lab"><font color="Green" title="E70">A2</font></span>: 
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:90_1"/><span class="lab"><font color="Green" title="E71">B2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E5:90_1"/><span class="lab"><font color="Green" title="E72">C2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">a</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_1"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="ref" href="lexbfs.html#D2" title="LEXBFS:def.2">LEXBFS:def 2</a></span>;<br/>
<a NAME="E6:90_1"/>
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_1"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/>
<a NAME="E7:90_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">D2</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:90"><span class="lab"><font color="Green" title="E56">S2d</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E8:90_1"/>
<font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_1"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/>
<a NAME="E9:90_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:90"><span class="lab"><font color="Green" title="E57">S2e</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E10:90_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:90_1"><span class="lab"><font color="Green" title="E71">B2</font></span></a>, <a class="txt" href="#E5:90_1"><span class="lab"><font color="Green" title="E72">C2</font></span></a>, <a class="txt" href="#E7:90_1"><span class="lab"><font color="Green" title="E73">D2</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:90_1"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E25:90"/><span class="lab"><font color="Green" title="E70">Y1b</font></span>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span></span>)</span> implies <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> ) &amp; ( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">R</font> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span></span>)</span> ) )</span><br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">R</font> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span></span>)</span> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:90_2_1_1"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:90_2_1_1"/><span class="lab"><font color="Green" title="E71">B2</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">Y</font>
 <span class="kw">and </span><br/><a NAME="E4:90_2_1_1"/><span class="lab"><font color="Green" title="E72">C2</font></span>: 
<font color="Maroon" title="c17">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c18">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:90_2_1_1"/><span class="lab"><font color="Green" title="E73">D2</font></span>: 
<font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:90_2_1_1"/><span class="lab"><font color="Green" title="E74">E2</font></span>: 
<font color="Maroon" title="c17">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:90_2_1_1"><span class="lab"><font color="Green" title="E72">C2</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E8:90_2_1_1"/><span class="lab"><font color="Green" title="E75">F2</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:90_2_1_1"><span class="lab"><font color="Green" title="E73">D2</font></span></a>, <a class="txt" href="#E7:90_2_1_1"><span class="lab"><font color="Green" title="E74">E2</font></span></a>, <a class="txt" href="#E3:90_2_1_1"><span class="lab"><font color="Green" title="E71">B2</font></span></a>, <a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="ref" href="lexbfs.html#D2" title="LEXBFS:def.2">LEXBFS:def 2</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_2_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:90_2_1_1_1"/>
( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font> or <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:90_2_1_1"><span class="lab"><font color="Green" title="E75">F2</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:90_2_1_1_1_1"/><span class="lab"><font color="Green" title="E76">S2</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font></span><br/><div class="add"><a NAME="E2:90_2_1_1_1_1"/>
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">U</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:90_2_1_1"><span class="lab"><font color="Green" title="E73">D2</font></span></a>, <a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/><a NAME="E3:90_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dilworth.html#K4" title="DILWORTH:func.4">Upper</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_2_1_1_1_1"><span class="lab"><font color="Green" title="E76">S2</font></span></a>, <a class="txt" href="#E5:90"><span class="lab"><font color="Green" title="E56">S2d</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:90_2_1_1_1_1"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="90_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:90_2_1_1_1_2"/><span class="lab"><font color="Green" title="E76">S2</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font></span><br/><div class="add"><a NAME="E2:90_2_1_1_1_2"/>
<font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:90_2_1_1"><span class="lab"><font color="Green" title="E73">D2</font></span></a>, <a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/><a NAME="E3:90_2_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dilworth.html#K3" title="DILWORTH:func.3">Lower</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_2_1_1_1_2"><span class="lab"><font color="Green" title="E76">S2</font></span></a>, <a class="txt" href="#E6:90"><span class="lab"><font color="Green" title="E57">S2e</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:90_2_1_1_1_2"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">assume </span><a NAME="E2:90_2_1"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span></span>)</span></span><br/><a NAME="E3:90_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A2</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="dilworth.html#K4" title="DILWORTH:func.4">Upper</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="dilworth.html#K3" title="DILWORTH:func.3">Lower</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="ref" href="dilworth.html#T23" target="_self" title="DILWORTH:th.23">ABunion</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_2_1_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:90_2_1_2"/>
( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dilworth.html#K4" title="DILWORTH:func.4">Upper</a> <font color="Maroon" title="c2">A</font> or <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dilworth.html#K3" title="DILWORTH:func.3">Lower</a> <font color="Maroon" title="c2">A</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_2_1"><span class="lab"><font color="Green" title="E71">A2</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_2_1_2_1"><span class="kw">suppose </span></a><a NAME="E1:90_2_1_2_1"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dilworth.html#K4" title="DILWORTH:func.4">Upper</a> <font color="Maroon" title="c2">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:90_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">U</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:90"><span class="lab"><font color="Green" title="E56">S2d</font></span></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:90_2_1_2_1"/><span class="lab"><font color="Green" title="E72">A3</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">Y</font>
 <span class="kw">and </span><br/><a NAME="E5:90_2_1_2_1"/><span class="lab"><font color="Green" title="E73">B3</font></span>: 
<font color="Maroon" title="c17">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">U</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c18">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:90_2_1_2_1"/><span class="lab"><font color="Green" title="E74">C3</font></span>: 
<font color="Maroon" title="c18">a</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="c11">f</font>
 <span class="kw">and </span><br/><a NAME="E8:90_2_1_2_1"/><span class="lab"><font color="Green" title="E75">D3</font></span>: 
<font color="Maroon" title="c17">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:90_2_1_2_1"><span class="lab"><font color="Green" title="E73">B3</font></span></a>, <a class="txt" href="#E14:90"><span class="lab"><font color="Green" title="E62">E1d</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E9:90_2_1_2_1"/><span class="lab"><font color="Green" title="E76">E3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:90_2_1_2_1"><span class="lab"><font color="Green" title="E74">C3</font></span></a>, <a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="txt" href="#E13:90"><span class="lab"><font color="Green" title="E61">E1b</font></span></a>, <a class="txt" href="#E21:90"><span class="lab"><font color="Green" title="E66">G1b</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><a NAME="E10:90_2_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:90_2_1_2_1"><span class="lab"><font color="Green" title="E74">C3</font></span></a>, <a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="txt" href="#E13:90"><span class="lab"><font color="Green" title="E61">E1b</font></span></a>, <a class="txt" href="#E21:90"><span class="lab"><font color="Green" title="E66">G1b</font></span></a>, <a class="ref" href="lexbfs.html#D2" title="LEXBFS:def.2">LEXBFS:def 2</a></span>;<br/><a NAME="E11:90_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:90_2_1_2_1"><span class="lab"><font color="Green" title="E72">A3</font></span></a>, <a class="txt" href="#E8:90_2_1_2_1"><span class="lab"><font color="Green" title="E75">D3</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E12:90_2_1_2_1"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:90_2_1_2_1"><span class="lab"><font color="Green" title="E76">E3</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="90_2_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:90_2_1_2_2"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dilworth.html#K3" title="DILWORTH:func.3">Lower</a> <font color="Maroon" title="c2">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:90_2_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:90"><span class="lab"><font color="Green" title="E57">S2e</font></span></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:90_2_1_2_2"/><span class="lab"><font color="Green" title="E72">A3</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">Y</font>
 <span class="kw">and </span><br/><a NAME="E5:90_2_1_2_2"/><span class="lab"><font color="Green" title="E73">B3</font></span>: 
<font color="Maroon" title="c17">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c18">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:90_2_1_2_2"/><span class="lab"><font color="Green" title="E74">C3</font></span>: 
<font color="Maroon" title="c18">a</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="c13">g</font>
 <span class="kw">and </span><br/><a NAME="E8:90_2_1_2_2"/><span class="lab"><font color="Green" title="E75">D3</font></span>: 
<font color="Maroon" title="c17">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:90_2_1_2_2"><span class="lab"><font color="Green" title="E73">B3</font></span></a>, <a class="txt" href="#E22:90"><span class="lab"><font color="Green" title="E67">G1d</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E9:90_2_1_2_2"/><span class="lab"><font color="Green" title="E76">E3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:90_2_1_2_2"><span class="lab"><font color="Green" title="E74">C3</font></span></a>, <a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="txt" href="#E13:90"><span class="lab"><font color="Green" title="E61">E1b</font></span></a>, <a class="txt" href="#E21:90"><span class="lab"><font color="Green" title="E66">G1b</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><a NAME="E10:90_2_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:90_2_1_2_2"><span class="lab"><font color="Green" title="E74">C3</font></span></a>, <a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="txt" href="#E13:90"><span class="lab"><font color="Green" title="E61">E1b</font></span></a>, <a class="txt" href="#E21:90"><span class="lab"><font color="Green" title="E66">G1b</font></span></a>, <a class="ref" href="lexbfs.html#D2" title="LEXBFS:def.2">LEXBFS:def 2</a></span>;<br/><a NAME="E11:90_2_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:90_2_1_2_2"><span class="lab"><font color="Green" title="E72">A3</font></span></a>, <a class="txt" href="#E8:90_2_1_2_2"><span class="lab"><font color="Green" title="E75">D3</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E12:90_2_1_2_2"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:90_2_1_2_2"><span class="lab"><font color="Green" title="E76">E3</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E2:90_2"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E71">Y1c</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="90_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">a</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span> implies ( <font color="Maroon" title="c16">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( for <font color="Olive" title="b1">b</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span> &amp; <font color="Maroon" title="c16">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">b</font> holds <br/>not <font color="Maroon" title="c16">a</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">b</font> ) ) )</span><br/><span class="kw">assume </span><a NAME="E1:90_3"/>
<font color="Maroon" title="c16">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( for <font color="Olive" title="b1">b</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span> &amp; <font color="Maroon" title="c16">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">b</font> holds <br/>not <font color="Maroon" title="c16">a</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">b</font> ) )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:90_3"/><span class="lab"><font color="Green" title="E72">B2</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:90_3"/><span class="lab"><font color="Green" title="E73">C2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E5:90_3"/><span class="lab"><font color="Green" title="E74">D2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_3"><span class="lab"><font color="Green" title="E72">B2</font></span></a>, <a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="ref" href="lexbfs.html#D2" title="LEXBFS:def.2">LEXBFS:def 2</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c18">cfx</font> = <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font>;<br/><span class="kw">set </span><font color="Maroon" title="c19">cgx</font> = <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font>;<br/><a NAME="E6:90_3"/><span class="lab"><font color="Green" title="E75">E2</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">U</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_3"><span class="lab"><font color="Green" title="E72">B2</font></span></a>, <a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">cfx</font> = <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c10">Pa</font> ;<br/><a NAME="E8:90_3"/><span class="lab"><font color="Green" title="E76">E2g</font></span>: 
<font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_3"><span class="lab"><font color="Green" title="E72">B2</font></span></a>, <a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c21">cgx</font> = <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c12">Pb</font> ;<br/><a NAME="E10:90_3"/>
<font color="Maroon" title="c20">cfx</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="#E6:90_3"><span class="lab"><font color="Green" title="E75">E2</font></span></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E11:90_3"/>
<font color="Maroon" title="c16">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="#E4:90_3"><span class="lab"><font color="Green" title="E73">C2</font></span></a>, <a class="txt" href="#E5:90_3"><span class="lab"><font color="Green" title="E74">D2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">b</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span> &amp; <font color="Maroon" title="c16">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">b</font> holds <br/>not <font color="Maroon" title="c16">a</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">b</font></span><br/><span class="kw">let </span><font color="Maroon" title="c22">b</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span> &amp; <font color="Maroon" title="c16">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c22">b</font> implies not <font color="Maroon" title="c16">a</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c22">b</font> )</span><br/><span class="kw">assume </span><a NAME="E12:90_3"/>
<font color="Maroon" title="c22">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c22">b</font> implies not <font color="Maroon" title="c16">a</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c22">b</font> )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E14:90_3"/><span class="lab"><font color="Green" title="E77">B2y</font></span>: 
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E15:90_3"/><span class="lab"><font color="Green" title="E78">C2y</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E16:90_3"/><span class="lab"><font color="Green" title="E79">D2y</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">y</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:90_3"><span class="lab"><font color="Green" title="E77">B2y</font></span></a>, <a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="ref" href="lexbfs.html#D2" title="LEXBFS:def.2">LEXBFS:def 2</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c24">cfy</font> = <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">y</font>;<br/><span class="kw">set </span><font color="Maroon" title="c25">cgy</font> = <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">y</font>;<br/><a NAME="E17:90_3"/><span class="lab"><font color="Green" title="E80">E2y</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">U</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:90_3"><span class="lab"><font color="Green" title="E77">B2y</font></span></a>, <a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c26">cfy</font> = <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">y</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c10">Pa</font> ;<br/><a NAME="E19:90_3"/><span class="lab"><font color="Green" title="E81">E2yg</font></span>: 
<font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:90_3"><span class="lab"><font color="Green" title="E77">B2y</font></span></a>, <a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c27">cgy</font> = <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">y</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c12">Pb</font> ;<br/><span class="kw">assume </span><a NAME="E21:90_3"/><span class="lab"><font color="Green" title="E82">FF2</font></span>: 
<font color="Maroon" title="c16">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c22">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c16">a</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c22">b</font></span><br/><a NAME="E22:90_3"/><span class="lab"><font color="Green" title="E83">F2a</font></span>: 
<font color="Maroon" title="c20">cfx</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c26">cfy</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:90"><span class="lab"><font color="Green" title="E59">E1</font></span></a>, <a class="txt" href="#E21:90_3"><span class="lab"><font color="Green" title="E82">FF2</font></span></a>, <a class="txt" href="#E4:90_3"><span class="lab"><font color="Green" title="E73">C2</font></span></a>, <a class="txt" href="#E15:90_3"><span class="lab"><font color="Green" title="E78">C2y</font></span></a>, <a class="txt" href="#E3:90_3"><span class="lab"><font color="Green" title="E72">B2</font></span></a>, <a class="txt" href="#E14:90_3"><span class="lab"><font color="Green" title="E77">B2y</font></span></a>, <a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="txt" href="#E13:90"><span class="lab"><font color="Green" title="E61">E1b</font></span></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></span>;<br/><a NAME="E23:90_3"/><span class="lab"><font color="Green" title="E84">F2g</font></span>: 
<font color="Maroon" title="c21">cgx</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c27">cgy</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:90"><span class="lab"><font color="Green" title="E64">G1</font></span></a>, <a class="txt" href="#E21:90_3"><span class="lab"><font color="Green" title="E82">FF2</font></span></a>, <a class="txt" href="#E4:90_3"><span class="lab"><font color="Green" title="E73">C2</font></span></a>, <a class="txt" href="#E15:90_3"><span class="lab"><font color="Green" title="E78">C2y</font></span></a>, <a class="txt" href="#E3:90_3"><span class="lab"><font color="Green" title="E72">B2</font></span></a>, <a class="txt" href="#E14:90_3"><span class="lab"><font color="Green" title="E77">B2y</font></span></a>, <a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="txt" href="#E21:90"><span class="lab"><font color="Green" title="E66">G1b</font></span></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></span>;<br/><span class="kw">assume </span><a NAME="E24:90_3"/>
<font color="Maroon" title="c16">a</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c22">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E25:90_3"/><span class="kw">then </span>
<font color="Maroon" title="c16">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c22">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c28">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E27:90_3"/><span class="lab"><font color="Green" title="E85">F2b</font></span>: 
<font color="Maroon" title="c28">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c22">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><a NAME="E28:90_3"/><span class="lab"><font color="Green" title="E86">G2</font></span>: 
<font color="Maroon" title="c28">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:90_3"><span class="lab"><font color="Green" title="E85">F2b</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E29:90_3"/><span class="lab"><font color="Green" title="E87">H2</font></span>: 
<font color="Maroon" title="c28">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:90_3"><span class="lab"><font color="Green" title="E85">F2b</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c29">cfz</font> = <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">z</font>;<br/><span class="kw">set </span><font color="Maroon" title="c30">cgz</font> = <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">z</font>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:90_3_1"/>
( ( <font color="Maroon" title="c28">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">cfx</font> &amp; <font color="Maroon" title="c28">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">cfy</font> ) or ( <font color="Maroon" title="c28">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">cfx</font> &amp; <font color="Maroon" title="c28">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">cgy</font> ) or ( <font color="Maroon" title="c28">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">cgx</font> &amp; <font color="Maroon" title="c28">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">cfy</font> ) or ( <font color="Maroon" title="c28">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">cgx</font> &amp; <font color="Maroon" title="c28">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">cgy</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:90_3"><span class="lab"><font color="Green" title="E86">G2</font></span></a>, <a class="txt" href="#E29:90_3"><span class="lab"><font color="Green" title="E87">H2</font></span></a>, <a class="txt" href="#E4:90_3"><span class="lab"><font color="Green" title="E73">C2</font></span></a>, <a class="txt" href="#E15:90_3"><span class="lab"><font color="Green" title="E78">C2y</font></span></a>, <a class="txt" href="#E5:90_3"><span class="lab"><font color="Green" title="E74">D2</font></span></a>, <a class="txt" href="#E16:90_3"><span class="lab"><font color="Green" title="E79">D2y</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:90_3_1_1"/>
( <font color="Maroon" title="c28">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">cfx</font> &amp; <font color="Maroon" title="c28">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">cfy</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:90_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c28">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">cfx</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c26">cfy</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:90_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">cfx</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c26">cfy</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E4:90_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:90_3"><span class="lab"><font color="Green" title="E75">E2</font></span></a>, <a class="txt" href="#E17:90_3"><span class="lab"><font color="Green" title="E80">E2y</font></span></a>, <a class="txt" href="#E22:90_3"><span class="lab"><font color="Green" title="E83">F2a</font></span></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="90_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:90_3_1_2"/><span class="lab"><font color="Green" title="E88">S1</font></span>: 
( <font color="Maroon" title="c28">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">cfx</font> &amp; <font color="Maroon" title="c28">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">cgy</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:90_3_1_2"/><span class="lab"><font color="Green" title="E89">C3</font></span>: 
<font color="Maroon" title="c28">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_3_1_2"><span class="lab"><font color="Green" title="E88">S1</font></span></a>, <a class="txt" href="#E5:90"><span class="lab"><font color="Green" title="E56">S2d</font></span></a>, <a class="txt" href="#E6:90"><span class="lab"><font color="Green" title="E57">S2e</font></span></a>, <a class="ref" href="dilworth.html#T22" target="_self" title="DILWORTH:th.22">ABAC0</a></span>;<br/><a NAME="E3:90_3_1_2"/><span class="lab"><font color="Green" title="E90">D3</font></span>: 
<font color="Maroon" title="c28">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_3_1_2"><span class="lab"><font color="Green" title="E88">S1</font></span></a>, <a class="txt" href="#E5:90"><span class="lab"><font color="Green" title="E56">S2d</font></span></a>, <a class="txt" href="#E6:90"><span class="lab"><font color="Green" title="E57">S2e</font></span></a>, <a class="ref" href="dilworth.html#T22" target="_self" title="DILWORTH:th.22">ABAC0</a>, <a class="txt" href="#E12:90"><span class="lab"><font color="Green" title="E60">E1a</font></span></a></span>;<br/><a NAME="E4:90_3_1_2"/><span class="lab"><font color="Green" title="E91">E3</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">U</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_3_1_2"><span class="lab"><font color="Green" title="E88">S1</font></span></a>, <a class="txt" href="#E5:90"><span class="lab"><font color="Green" title="E56">S2d</font></span></a>, <a class="txt" href="#E6:90"><span class="lab"><font color="Green" title="E57">S2e</font></span></a>, <a class="ref" href="dilworth.html#T22" target="_self" title="DILWORTH:th.22">ABAC0</a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c31">cfz</font> = <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">z</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c10">Pa</font> ;<br/><a NAME="E6:90_3_1_2"/>
<font color="Maroon" title="c28">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">cfx</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c31">cfz</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_3_1_2"><span class="lab"><font color="Green" title="E88">S1</font></span></a>, <a class="txt" href="#E3:90_3_1_2"><span class="lab"><font color="Green" title="E90">D3</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E7:90_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c31">cfz</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c20">cfx</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><a NAME="E8:90_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c31">cfz</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">cfx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:90_3"><span class="lab"><font color="Green" title="E75">E2</font></span></a>, <a class="txt" href="#E4:90_3_1_2"><span class="lab"><font color="Green" title="E91">E3</font></span></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></span>;<br/><a NAME="E9:90_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">F3</font></span>: 
<font color="Maroon" title="c28">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:90"><span class="lab"><font color="Green" title="E59">E1</font></span></a>, <a class="txt" href="#E3:90_3"><span class="lab"><font color="Green" title="E72">B2</font></span></a>, <a class="txt" href="#E2:90_3_1_2"><span class="lab"><font color="Green" title="E89">C3</font></span></a>, <a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="txt" href="#E13:90"><span class="lab"><font color="Green" title="E61">E1b</font></span></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></span>;<br/><a NAME="E10:90_3_1_2"/><span class="lab"><font color="Green" title="E93">D3g</font></span>: 
<font color="Maroon" title="c28">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_3_1_2"><span class="lab"><font color="Green" title="E88">S1</font></span></a>, <a class="txt" href="#E5:90"><span class="lab"><font color="Green" title="E56">S2d</font></span></a>, <a class="txt" href="#E6:90"><span class="lab"><font color="Green" title="E57">S2e</font></span></a>, <a class="ref" href="dilworth.html#T22" target="_self" title="DILWORTH:th.22">ABAC0</a>, <a class="txt" href="#E20:90"><span class="lab"><font color="Green" title="E65">G1a</font></span></a></span>;<br/><a NAME="E11:90_3_1_2"/><span class="lab"><font color="Green" title="E94">E3g</font></span>: 
<font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_3_1_2"><span class="lab"><font color="Green" title="E88">S1</font></span></a>, <a class="txt" href="#E5:90"><span class="lab"><font color="Green" title="E56">S2d</font></span></a>, <a class="txt" href="#E6:90"><span class="lab"><font color="Green" title="E57">S2e</font></span></a>, <a class="ref" href="dilworth.html#T22" target="_self" title="DILWORTH:th.22">ABAC0</a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c32">cgz</font> = <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">z</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c12">Pb</font> ;<br/><a NAME="E13:90_3_1_2"/>
<font color="Maroon" title="c28">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c32">cgz</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c27">cgy</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_3_1_2"><span class="lab"><font color="Green" title="E88">S1</font></span></a>, <a class="txt" href="#E10:90_3_1_2"><span class="lab"><font color="Green" title="E93">D3g</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E14:90_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c32">cgz</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c27">cgy</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><a NAME="E15:90_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c32">cgz</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">cgy</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:90_3"><span class="lab"><font color="Green" title="E81">E2yg</font></span></a>, <a class="txt" href="#E11:90_3_1_2"><span class="lab"><font color="Green" title="E94">E3g</font></span></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E16:90_3_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:90_3_1_2"><span class="lab"><font color="Green" title="E92">F3</font></span></a>, <a class="txt" href="#E21:90_3"><span class="lab"><font color="Green" title="E82">FF2</font></span></a>, <a class="txt" href="#E4:90_3"><span class="lab"><font color="Green" title="E73">C2</font></span></a>, <a class="txt" href="#E15:90_3"><span class="lab"><font color="Green" title="E78">C2y</font></span></a>, <a class="txt" href="#E19:90"><span class="lab"><font color="Green" title="E64">G1</font></span></a>, <a class="txt" href="#E14:90_3"><span class="lab"><font color="Green" title="E77">B2y</font></span></a>, <a class="txt" href="#E2:90_3_1_2"><span class="lab"><font color="Green" title="E89">C3</font></span></a>, <a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="txt" href="#E21:90"><span class="lab"><font color="Green" title="E66">G1b</font></span></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="90_3_1_3"><span class="kw">suppose </span></a><a NAME="E1:90_3_1_3"/><span class="lab"><font color="Green" title="E88">S1</font></span>: 
( <font color="Maroon" title="c28">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">cgx</font> &amp; <font color="Maroon" title="c28">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">cfy</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:90_3_1_3"/><span class="lab"><font color="Green" title="E89">C3</font></span>: 
<font color="Maroon" title="c28">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_3_1_3"><span class="lab"><font color="Green" title="E88">S1</font></span></a>, <a class="txt" href="#E5:90"><span class="lab"><font color="Green" title="E56">S2d</font></span></a>, <a class="txt" href="#E6:90"><span class="lab"><font color="Green" title="E57">S2e</font></span></a>, <a class="ref" href="dilworth.html#T22" target="_self" title="DILWORTH:th.22">ABAC0</a></span>;<br/><a NAME="E3:90_3_1_3"/><span class="lab"><font color="Green" title="E90">D3</font></span>: 
<font color="Maroon" title="c28">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_3_1_3"><span class="lab"><font color="Green" title="E88">S1</font></span></a>, <a class="txt" href="#E5:90"><span class="lab"><font color="Green" title="E56">S2d</font></span></a>, <a class="txt" href="#E6:90"><span class="lab"><font color="Green" title="E57">S2e</font></span></a>, <a class="ref" href="dilworth.html#T22" target="_self" title="DILWORTH:th.22">ABAC0</a>, <a class="txt" href="#E12:90"><span class="lab"><font color="Green" title="E60">E1a</font></span></a></span>;<br/><a NAME="E4:90_3_1_3"/><span class="lab"><font color="Green" title="E91">E3</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">U</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_3_1_3"><span class="lab"><font color="Green" title="E88">S1</font></span></a>, <a class="txt" href="#E5:90"><span class="lab"><font color="Green" title="E56">S2d</font></span></a>, <a class="txt" href="#E6:90"><span class="lab"><font color="Green" title="E57">S2e</font></span></a>, <a class="ref" href="dilworth.html#T22" target="_self" title="DILWORTH:th.22">ABAC0</a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c31">cfz</font> = <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">z</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c10">Pa</font> ;<br/><a NAME="E6:90_3_1_3"/>
<font color="Maroon" title="c28">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">cfy</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c31">cfz</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_3_1_3"><span class="lab"><font color="Green" title="E88">S1</font></span></a>, <a class="txt" href="#E3:90_3_1_3"><span class="lab"><font color="Green" title="E90">D3</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E7:90_3_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c31">cfz</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c26">cfy</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><a NAME="E8:90_3_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c31">cfz</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">cfy</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:90_3"><span class="lab"><font color="Green" title="E80">E2y</font></span></a>, <a class="txt" href="#E4:90_3_1_3"><span class="lab"><font color="Green" title="E91">E3</font></span></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></span>;<br/><a NAME="E9:90_3_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">F3</font></span>: 
<font color="Maroon" title="c28">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:90"><span class="lab"><font color="Green" title="E59">E1</font></span></a>, <a class="txt" href="#E14:90_3"><span class="lab"><font color="Green" title="E77">B2y</font></span></a>, <a class="txt" href="#E2:90_3_1_3"><span class="lab"><font color="Green" title="E89">C3</font></span></a>, <a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="txt" href="#E13:90"><span class="lab"><font color="Green" title="E61">E1b</font></span></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></span>;<br/><a NAME="E10:90_3_1_3"/><span class="lab"><font color="Green" title="E93">D3g</font></span>: 
<font color="Maroon" title="c28">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_3_1_3"><span class="lab"><font color="Green" title="E88">S1</font></span></a>, <a class="txt" href="#E5:90"><span class="lab"><font color="Green" title="E56">S2d</font></span></a>, <a class="txt" href="#E6:90"><span class="lab"><font color="Green" title="E57">S2e</font></span></a>, <a class="ref" href="dilworth.html#T22" target="_self" title="DILWORTH:th.22">ABAC0</a>, <a class="txt" href="#E20:90"><span class="lab"><font color="Green" title="E65">G1a</font></span></a></span>;<br/><a NAME="E11:90_3_1_3"/><span class="lab"><font color="Green" title="E94">E3g</font></span>: 
<font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_3_1_3"><span class="lab"><font color="Green" title="E88">S1</font></span></a>, <a class="txt" href="#E5:90"><span class="lab"><font color="Green" title="E56">S2d</font></span></a>, <a class="txt" href="#E6:90"><span class="lab"><font color="Green" title="E57">S2e</font></span></a>, <a class="ref" href="dilworth.html#T22" target="_self" title="DILWORTH:th.22">ABAC0</a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c32">cgz</font> = <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">z</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c12">Pb</font> ;<br/><a NAME="E13:90_3_1_3"/>
<font color="Maroon" title="c28">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c32">cgz</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c21">cgx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_3_1_3"><span class="lab"><font color="Green" title="E88">S1</font></span></a>, <a class="txt" href="#E10:90_3_1_3"><span class="lab"><font color="Green" title="E93">D3g</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E14:90_3_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c32">cgz</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c21">cgx</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><a NAME="E15:90_3_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c32">cgz</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">cgx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:90_3"><span class="lab"><font color="Green" title="E76">E2g</font></span></a>, <a class="txt" href="#E11:90_3_1_3"><span class="lab"><font color="Green" title="E94">E3g</font></span></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E16:90_3_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:90_3_1_3"><span class="lab"><font color="Green" title="E92">F3</font></span></a>, <a class="txt" href="#E21:90_3"><span class="lab"><font color="Green" title="E82">FF2</font></span></a>, <a class="txt" href="#E4:90_3"><span class="lab"><font color="Green" title="E73">C2</font></span></a>, <a class="txt" href="#E15:90_3"><span class="lab"><font color="Green" title="E78">C2y</font></span></a>, <a class="txt" href="#E19:90"><span class="lab"><font color="Green" title="E64">G1</font></span></a>, <a class="txt" href="#E3:90_3"><span class="lab"><font color="Green" title="E72">B2</font></span></a>, <a class="txt" href="#E2:90_3_1_3"><span class="lab"><font color="Green" title="E89">C3</font></span></a>, <a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="txt" href="#E21:90"><span class="lab"><font color="Green" title="E66">G1b</font></span></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="90_3_1_4"><span class="kw">suppose </span></a><a NAME="E1:90_3_1_4"/>
( <font color="Maroon" title="c28">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">cgx</font> &amp; <font color="Maroon" title="c28">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">cgy</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:90_3_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c28">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">cgx</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c27">cgy</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:90_3_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c21">cgx</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c27">cgy</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E4:90_3_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:90_3"><span class="lab"><font color="Green" title="E76">E2g</font></span></a>, <a class="txt" href="#E19:90_3"><span class="lab"><font color="Green" title="E81">E2yg</font></span></a>, <a class="txt" href="#E23:90_3"><span class="lab"><font color="Green" title="E84">F2g</font></span></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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>
<a NAME="E27:90"/><span class="lab"><font color="Green" title="E72">Z1</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span> holds <br/><font color="Olive" title="b1">x</font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <font color="Maroon" title="c1">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span> implies <font color="Maroon" title="c16">c</font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <font color="Maroon" title="c1">R</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:90_4"/>
<font color="Maroon" title="c16">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">c</font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <font color="Maroon" title="c1">R</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:90_4"/><span class="lab"><font color="Green" title="E73">B2</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:90_4"/><span class="lab"><font color="Green" title="E74">C2</font></span>: 
<font color="Maroon" title="c16">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E5:90_4"/><span class="lab"><font color="Green" title="E75">D2</font></span>: 
<font color="Maroon" title="c16">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_4"><span class="lab"><font color="Green" title="E73">B2</font></span></a>, <a class="txt" href="#E4:90_4"><span class="lab"><font color="Green" title="E74">C2</font></span></a>, <a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="ref" href="lexbfs.html#D2" title="LEXBFS:def.2">LEXBFS:def 2</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c18">cf</font> = <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c19">cg</font> = <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font>;<br/>
<a NAME="E6:90_4"/>
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_4"><span class="lab"><font color="Green" title="E73">B2</font></span></a>, <a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/>
<a NAME="E7:90_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">G2b</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <font color="Maroon" title="c10">Pa</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#D11" target="_self" title="DILWORTH:def.11">LCpart</a></span>;<br/>
<a NAME="E8:90_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">G2a</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#T28" target="_self" title="DILWORTH:th.28">SPClique</a></span>;<br/>
<a NAME="E9:90_4"/>
<font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_4"><span class="lab"><font color="Green" title="E73">B2</font></span></a>, <a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/>
<a NAME="E10:90_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">H2b</font></span>: 
<font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <font color="Maroon" title="c12">Pb</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#D11" target="_self" title="DILWORTH:def.11">LCpart</a></span>;<br/>
<a NAME="E11:90_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">H2a</font></span>: 
<font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#T28" target="_self" title="DILWORTH:th.28">SPClique</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">c9</font> = <font color="Maroon" title="c16">c</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:90_4"><span class="lab"><font color="Green" title="E75">D2</font></span></a>, <a class="txt" href="#E8:90_4"><span class="lab"><font color="Green" title="E77">G2a</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c21">a</font>, <font color="Maroon" title="c22">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">c9</font> &amp; <font color="Maroon" title="c22">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">c9</font> &amp; <font color="Maroon" title="c21">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c22">b</font> &amp; not <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> implies <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:90_4_1"/><span class="lab"><font color="Green" title="E80">A3</font></span>: 
<font color="Maroon" title="c21">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">c9</font>
 <span class="kw">and </span><br/><a NAME="E2:90_4_1"/><span class="lab"><font color="Green" title="E81">B3</font></span>: 
<font color="Maroon" title="c22">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">c9</font>
 <span class="kw">and </span><br/><a NAME="E3:90_4_1"/><span class="lab"><font color="Green" title="E82">C3</font></span>: 
<font color="Maroon" title="c21">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c22">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><a NAME="E4:90_4_1"/><span class="lab"><font color="Green" title="E83">E3</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_4"><span class="lab"><font color="Green" title="E73">B2</font></span></a>, <a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="txt" href="#E12:90"><span class="lab"><font color="Green" title="E60">E1a</font></span></a></span>;<br/><a NAME="E5:90_4_1"/><span class="lab"><font color="Green" title="E84">F3</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_4"><span class="lab"><font color="Green" title="E73">B2</font></span></a>, <a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="txt" href="#E20:90"><span class="lab"><font color="Green" title="E65">G1a</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c23">x</font> = <font color="Maroon" title="c17">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_4"><span class="lab"><font color="Green" title="E73">B2</font></span></a>, <a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="txt" href="#E13:90"><span class="lab"><font color="Green" title="E61">E1b</font></span></a>, <a class="txt" href="#E21:90"><span class="lab"><font color="Green" title="E66">G1b</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_4_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:90_4_1_1"/>
( ( <font color="Maroon" title="c21">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> &amp; <font color="Maroon" title="c22">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> ) or ( <font color="Maroon" title="c21">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> &amp; <font color="Maroon" title="c22">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> ) or ( <font color="Maroon" title="c21">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> &amp; <font color="Maroon" title="c22">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> ) or ( <font color="Maroon" title="c21">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> &amp; <font color="Maroon" title="c22">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_4_1"><span class="lab"><font color="Green" title="E80">A3</font></span></a>, <a class="txt" href="#E2:90_4_1"><span class="lab"><font color="Green" title="E81">B3</font></span></a>, <a class="txt" href="#E5:90_4"><span class="lab"><font color="Green" title="E75">D2</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_4_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:90_4_1_1_1"/>
( <font color="Maroon" title="c21">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> &amp; <font color="Maroon" title="c22">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:90_4_1_1_1"/>
( <font color="Maroon" title="c21">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c22">b</font> or <font color="Maroon" title="c22">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c21">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:90_4"><span class="lab"><font color="Green" title="E77">G2a</font></span></a>, <a class="txt" href="#E3:90_4_1"><span class="lab"><font color="Green" title="E82">C3</font></span></a>, <a class="ref" href="dilworth.html#T6" target="_self" title="DILWORTH:th.6">DClique</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="90_4_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:90_4_1_1_2"/><span class="lab"><font color="Green" title="E85">S1</font></span>: 
( <font color="Maroon" title="c21">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> &amp; <font color="Maroon" title="c22">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><div class="add"><a NAME="E2:90_4_1_1_2"/><span class="lab"><font color="Green" title="E86">A4</font></span>: 
( <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">a</font> or <font color="Maroon" title="c23">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c21">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:90_4_1"><span class="lab"><font color="Green" title="E83">E3</font></span></a>, <a class="txt" href="#E1:90_4_1_1_2"><span class="lab"><font color="Green" title="E85">S1</font></span></a>, <a class="txt" href="#E3:90_4"><span class="lab"><font color="Green" title="E73">B2</font></span></a>, <a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="txt" href="#E7:90_4"><span class="lab"><font color="Green" title="E76">G2b</font></span></a>, <a class="ref" href="dilworth.html#T35" target="_self" title="DILWORTH:th.35">PabSmin</a></span>;<br/><a NAME="E3:90_4_1_1_2"/><span class="lab"><font color="Green" title="E87">B4</font></span>: 
( <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">b</font> or <font color="Maroon" title="c22">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c23">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:90_4_1"><span class="lab"><font color="Green" title="E84">F3</font></span></a>, <a class="txt" href="#E1:90_4_1_1_2"><span class="lab"><font color="Green" title="E85">S1</font></span></a>, <a class="txt" href="#E3:90_4"><span class="lab"><font color="Green" title="E73">B2</font></span></a>, <a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="txt" href="#E10:90_4"><span class="lab"><font color="Green" title="E78">H2b</font></span></a>, <a class="ref" href="dilworth.html#T34" target="_self" title="DILWORTH:th.34">PbeSmax</a></span>;<br/><span class="kw">thus </span><a NAME="E4:90_4_1_1_2"/>
( <font color="Maroon" title="c21">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c22">b</font> or <font color="Maroon" title="c22">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c21">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_4_1"><span class="lab"><font color="Green" title="E82">C3</font></span></a>, <a class="txt" href="#E2:90_4_1_1_2"><span class="lab"><font color="Green" title="E86">A4</font></span></a>, <a class="txt" href="#E3:90_4_1_1_2"><span class="lab"><font color="Green" title="E87">B4</font></span></a>, <a class="ref" href="yellow_0.html#D2" title="YELLOW_0:def.2">YELLOW_0:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="90_4_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:90_4_1_1_3"/><span class="lab"><font color="Green" title="E85">S1</font></span>: 
( <font color="Maroon" title="c21">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> &amp; <font color="Maroon" title="c22">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><div class="add"><a NAME="E2:90_4_1_1_3"/><span class="lab"><font color="Green" title="E86">A4</font></span>: 
( <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">a</font> or <font color="Maroon" title="c21">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c23">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:90_4_1"><span class="lab"><font color="Green" title="E84">F3</font></span></a>, <a class="txt" href="#E1:90_4_1_1_3"><span class="lab"><font color="Green" title="E85">S1</font></span></a>, <a class="txt" href="#E3:90_4"><span class="lab"><font color="Green" title="E73">B2</font></span></a>, <a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="txt" href="#E10:90_4"><span class="lab"><font color="Green" title="E78">H2b</font></span></a>, <a class="ref" href="dilworth.html#T34" target="_self" title="DILWORTH:th.34">PbeSmax</a></span>;<br/><a NAME="E3:90_4_1_1_3"/><span class="lab"><font color="Green" title="E87">B4</font></span>: 
( <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">b</font> or <font color="Maroon" title="c23">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c22">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:90_4_1"><span class="lab"><font color="Green" title="E83">E3</font></span></a>, <a class="txt" href="#E1:90_4_1_1_3"><span class="lab"><font color="Green" title="E85">S1</font></span></a>, <a class="txt" href="#E3:90_4"><span class="lab"><font color="Green" title="E73">B2</font></span></a>, <a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="txt" href="#E7:90_4"><span class="lab"><font color="Green" title="E76">G2b</font></span></a>, <a class="ref" href="dilworth.html#T35" target="_self" title="DILWORTH:th.35">PabSmin</a></span>;<br/><span class="kw">thus </span><a NAME="E4:90_4_1_1_3"/>
( <font color="Maroon" title="c21">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c22">b</font> or <font color="Maroon" title="c22">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c21">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_4_1"><span class="lab"><font color="Green" title="E82">C3</font></span></a>, <a class="txt" href="#E2:90_4_1_1_3"><span class="lab"><font color="Green" title="E86">A4</font></span></a>, <a class="txt" href="#E3:90_4_1_1_3"><span class="lab"><font color="Green" title="E87">B4</font></span></a>, <a class="ref" href="yellow_0.html#D2" title="YELLOW_0:def.2">YELLOW_0:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="90_4_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:90_4_1_1_4"/>
( <font color="Maroon" title="c21">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> &amp; <font color="Maroon" title="c22">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:90_4_1_1_4"/>
( <font color="Maroon" title="c21">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c22">b</font> or <font color="Maroon" title="c22">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c21">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:90_4"><span class="lab"><font color="Green" title="E79">H2a</font></span></a>, <a class="txt" href="#E3:90_4_1"><span class="lab"><font color="Green" title="E82">C3</font></span></a>, <a class="ref" href="dilworth.html#T6" target="_self" title="DILWORTH:th.6">DClique</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E14:90_4"/>
<font color="Maroon" title="c16">c</font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#T6" target="_self" title="DILWORTH:th.6">DClique</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E28:90"/><span class="lab"><font color="Green" title="E73">H1</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">x1</font>, <font color="Maroon" title="c17">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span> or  not <font color="Maroon" title="c17">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x2</font> or <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:90_5"/><span class="lab"><font color="Green" title="E74">A2</font></span>: 
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:90_5"/><span class="lab"><font color="Green" title="E75">B2</font></span>: 
<font color="Maroon" title="c17">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:90_5"/><span class="lab"><font color="Green" title="E76">C2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x2</font></span><br/>

<a NAME="E4:90_5"/><span class="lab"><font color="Green" title="E77">D2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:90"><span class="lab"><font color="Green" title="E72">Z1</font></span></a>, <a class="txt" href="#E1:90_5"><span class="lab"><font color="Green" title="E74">A2</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<a NAME="E5:90_5"/><span class="lab"><font color="Green" title="E78">D2b</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="txt" href="#E1:90_5"><span class="lab"><font color="Green" title="E74">A2</font></span></a>, <a class="ref" href="lexbfs.html#D2" title="LEXBFS:def.2">LEXBFS:def 2</a></span>;<br/>
<a NAME="E6:90_5"/>
( <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font> &amp; <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_5"><span class="lab"><font color="Green" title="E74">A2</font></span></a>, <a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="txt" href="#E12:90"><span class="lab"><font color="Green" title="E60">E1a</font></span></a>, <a class="txt" href="#E20:90"><span class="lab"><font color="Green" title="E65">G1a</font></span></a></span>;<br/>
<a NAME="E7:90_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">D2a</font></span>: 
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:90_5"><span class="lab"><font color="Green" title="E78">D2b</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E8:90_5"/><span class="lab"><font color="Green" title="E80">E2b</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="txt" href="#E2:90_5"><span class="lab"><font color="Green" title="E75">B2</font></span></a>, <a class="ref" href="lexbfs.html#D2" title="LEXBFS:def.2">LEXBFS:def 2</a></span>;<br/>
<a NAME="E9:90_5"/>
( <font color="Maroon" title="c17">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x2</font> &amp; <font color="Maroon" title="c17">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:90_5"><span class="lab"><font color="Green" title="E75">B2</font></span></a>, <a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="txt" href="#E12:90"><span class="lab"><font color="Green" title="E60">E1a</font></span></a>, <a class="txt" href="#E20:90"><span class="lab"><font color="Green" title="E65">G1a</font></span></a></span>;<br/>
<a NAME="E10:90_5"/><span class="kw">then </span>
<font color="Maroon" title="c17">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:90_5"><span class="lab"><font color="Green" title="E80">E2b</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:90_5"/>
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="txt" href="#E1:90_5"><span class="lab"><font color="Green" title="E74">A2</font></span></a>, <a class="txt" href="#E2:90_5"><span class="lab"><font color="Green" title="E75">B2</font></span></a>, <a class="txt" href="#E3:90_5"><span class="lab"><font color="Green" title="E76">C2</font></span></a>, <a class="txt" href="#E4:90_5"><span class="lab"><font color="Green" title="E77">D2</font></span></a>, <a class="txt" href="#E7:90_5"><span class="lab"><font color="Green" title="E79">D2a</font></span></a>, <a class="ref" href="dilworth.html#T15" target="_self" title="DILWORTH:th.15">ACClique</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c16">C</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span> as   <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c1">R</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:90"><span class="lab"><font color="Green" title="E69">Y1a</font></span></a>, <a class="txt" href="#E25:90"><span class="lab"><font color="Green" title="E70">Y1b</font></span></a>, <a class="txt" href="#E26:90"><span class="lab"><font color="Green" title="E71">Y1c</font></span></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a>, <a class="txt" href="#E27:90"><span class="lab"><font color="Green" title="E72">Z1</font></span></a>, <a class="ref" href="dilworth.html#D11" target="_self" title="DILWORTH:def.11">LCpart</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c16">C</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c16">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c1">R</font></span><br/>

 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c16">C</font> = 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="lexbfs.html#K1" title="LEXBFS:func.1">.\/</a> <font color="Maroon" title="c13">g</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:90"><span class="lab"><font color="Green" title="E73">H1</font></span></a>, <a class="ref" href="pre_poly.html#T19" title="PRE_POLY:th.19">PRE_POLY:19</a></span>
<br/>.= 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:90"><span class="lab"><font color="Green" title="E68">I1a</font></span></a>, <a class="txt" href="#E13:90"><span class="lab"><font color="Green" title="E61">E1b</font></span></a>, <a class="txt" href="#E21:90"><span class="lab"><font color="Green" title="E66">G1b</font></span></a>, <a class="ref" href="card_1.html#T104" title="CARD_1:th.104">CARD_1:104</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E31:90"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c16">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90"><span class="lab"><font color="Green" title="E52">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
