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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="wellfnd1.html#V1" title="WELLFND1:attr.1">well_founded</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">V</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">H</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<span class="p2">(<span class="default"><a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Olive" title="b1">V</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Olive" title="b1">V</font><br/> for <font color="Olive" title="b3">F1</font>, <font color="Olive" title="b4">F2</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Olive" title="b1">V</font>  st <font color="Olive" title="b3">F1</font> <a href="wellfnd1.html#R1" title="WELLFND1:pred.1">is_recursively_expressed_by</a> <font color="Olive" title="b2">H</font> &amp; <font color="Olive" title="b4">F2</font> <a href="wellfnd1.html#R1" title="WELLFND1:pred.1">is_recursively_expressed_by</a> <font color="Olive" title="b2">H</font> holds <br/><font color="Olive" title="b3">F1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b4">F2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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"> for <font color="Olive" title="b1">H</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<span class="p2">(<span class="default"><a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">V</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c2">V</font><br/> for <font color="Olive" title="b2">F1</font>, <font color="Olive" title="b3">F2</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">V</font>  st <font color="Olive" title="b2">F1</font> <a href="wellfnd1.html#R1" title="WELLFND1:pred.1">is_recursively_expressed_by</a> <font color="Olive" title="b1">H</font> &amp; <font color="Olive" title="b3">F2</font> <a href="wellfnd1.html#R1" title="WELLFND1:pred.1">is_recursively_expressed_by</a> <font color="Olive" title="b1">H</font> holds <br/><font color="Olive" title="b2">F1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b3">F2</font></span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">c</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="c4">r</font> = the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">H</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<span class="p2">(<span class="default"><a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">V</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c2">V</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">F1</font>, <font color="Olive" title="b2">F2</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">V</font>  st <font color="Olive" title="b1">F1</font> <a href="wellfnd1.html#R1" title="WELLFND1:pred.1">is_recursively_expressed_by</a> <font color="Maroon" title="c5">H</font> &amp; <font color="Olive" title="b2">F2</font> <a href="wellfnd1.html#R1" title="WELLFND1:pred.1">is_recursively_expressed_by</a> <font color="Maroon" title="c5">H</font> holds <br/><font color="Olive" title="b1">F1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">F2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">F1</font>, <font color="Maroon" title="c7">F2</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">V</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="c6">F1</font> <a href="wellfnd1.html#R1" title="WELLFND1:pred.1">is_recursively_expressed_by</a> <font color="Maroon" title="c5">H</font> &amp; <font color="Maroon" title="c7">F2</font> <a href="wellfnd1.html#R1" title="WELLFND1:pred.1">is_recursively_expressed_by</a> <font color="Maroon" title="c5">H</font> implies <font color="Maroon" title="c6">F1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">F2</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:27"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
<font color="Maroon" title="c6">F1</font> <a href="wellfnd1.html#R1" title="WELLFND1:pred.1">is_recursively_expressed_by</a> <font color="Maroon" title="c5">H</font>
 <span class="kw">and </span><br/><a NAME="E2:27"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
<font color="Maroon" title="c7">F2</font> <a href="wellfnd1.html#R1" title="WELLFND1:pred.1">is_recursively_expressed_by</a> <font color="Maroon" title="c5">H</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="c6">F1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">F2</font></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span><font color="Maroon" title="c6">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<a NAME="E3:27"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">F2</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 <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:27"/>
<font color="Maroon" title="c6">F1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">F2</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/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 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"> such that </span><br/><a NAME="E6:27"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
<font color="Maroon" title="c6">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c8">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> ;<br/>
<a NAME="E8:27"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<font color="Maroon" title="c1">R</font> is  <a href="wellfnd1.html#V1" title="WELLFND1:attr.1">well_founded</a> 
 
;<br/>
<a NAME="E9:27"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">x</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:27"><span class="lab"><font color="Green" title="E19">A4</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">x0</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E11:27"/><span class="lab"><font color="Green" title="E22">A7</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">x0</font>]
 <span class="kw">and </span><br/><a NAME="E12:27"/><span class="lab"><font color="Green" title="E23">A8</font></span>: 
for <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>  holds <br/>(  not <font color="Maroon" title="c10">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">y</font> or  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">y</font>] or  not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c10">x0</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="wellfnd1.html#S2" target="_self" title="WELLFND1:sch.2">WELLFND1:sch 2</a>(<a class="txt" href="#E9:27"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E8:27"><span class="lab"><font color="Green" title="E20">A5</font></span></a>);<br/></span>
<a NAME="E13:27"/><span class="lab"><font color="Green" title="E24">A9</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">F1</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 <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E14:27"/>
<font color="Maroon" title="c6">F1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">x0</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c7">F2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">x0</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c11">fr</font> = <font color="Maroon" title="c6">F1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">x0</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">gr</font> = <font color="Maroon" title="c7">F2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">x0</font></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E1:27_1"/><span class="lab"><font color="Green" title="E25">A10</font></span>: 
not <font color="Maroon" title="c6">F1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">x0</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c7">F2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">x0</font></span>)</span>
 ; <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="E2:27_1"/><span class="lab"><font color="Green" title="E26">A11</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">x0</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">x0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:27"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="ref" href="wellfnd1.html#T4" target="_self" title="WELLFND1:th.4">Th4</a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/>
<a NAME="E3:27_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">x0</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">x0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="wellfnd1.html#T4" target="_self" title="WELLFND1:th.4">Th4</a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:27_1"/><span class="lab"><font color="Green" title="E27">A12</font></span>: 
<font color="Maroon" title="c13">x1</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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">x0</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:27_1"/><span class="lab"><font color="Green" title="E28">A13</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">F1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">x0</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">x0</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_1"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="txt" href="#E1:27_1"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></span>;<br/>
<a NAME="E7:27_1"/><span class="lab"><font color="Green" title="E29">A14</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c10">x0</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_1"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="txt" href="#E5:27_1"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">x1</font> = <font color="Maroon" title="c13">x1</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="#E5:27_1"><span class="lab"><font color="Green" title="E27">A12</font></span></a></span>;<br/>
<a NAME="E9:27_1"/><span class="lab"><font color="Green" title="E30">A15</font></span>: 
<font color="Maroon" title="c14">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">x0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_1"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="txt" href="#E5:27_1"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></span>;<br/>
<a NAME="E10:27_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">F1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">x0</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">x0</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_1"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="txt" href="#E5:27_1"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:27_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:27"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E6:27_1"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E7:27_1"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="txt" href="#E9:27_1"><span class="lab"><font color="Green" title="E30">A15</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><span class="kw">end;</span></div>
<span class="kw">then </span><font color="Maroon" title="c6">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x0</font> = 
<font color="Maroon" title="c5">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">x0</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">F2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="wellfnd1.html#D5" target="_self" title="WELLFND1:def.5">Def5</a></span>
<br/>.= 
<font color="Maroon" title="c7">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="wellfnd1.html#D5" target="_self" title="WELLFND1:def.5">Def5</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E16:27"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:27"><span class="lab"><font color="Green" title="E22">A7</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>
