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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a>  <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</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">f</font> being  <a href="quantal1.html#V14" title="QUANTAL1:attr.14">monotone</a>  <a href="quantal1.html#NM1" title="QUANTAL1:NM.1">UnOp</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">a</font> holds <br/>for <font color="Olive" title="b3">O1</font>, <font color="Olive" title="b4">O2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b3">O1</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Olive" title="b4">O2</font> &amp; not <font color="Olive" title="b1">f</font>,<font color="Olive" title="b4">O2</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Olive" title="b2">a</font> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a> <font color="Olive" title="b1">f</font> holds <br/><font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">O1</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b4">O2</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Olive" title="b2">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="quantal1.html#V14" title="QUANTAL1:attr.14">monotone</a>  <a href="quantal1.html#NM1" title="QUANTAL1:NM.1">UnOp</a> of <font color="Maroon" title="c1">L</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">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">a</font> holds <br/>for <font color="Olive" title="b2">O1</font>, <font color="Olive" title="b3">O2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b2">O1</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Olive" title="b3">O2</font> &amp; not <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b3">O2</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Olive" title="b1">a</font> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a> <font color="Maroon" title="c2">f</font> holds <br/><font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">O1</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b3">O2</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Olive" title="b1">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</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="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">a</font> implies for <font color="Olive" title="b1">O1</font>, <font color="Olive" title="b2">O2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O1</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Olive" title="b2">O2</font> &amp; not <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">O2</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c3">a</font> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a> <font color="Maroon" title="c2">f</font> holds <br/><font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">O1</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">O2</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c3">a</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:41"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">a</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">O1</font>, <font color="Olive" title="b2">O2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O1</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Olive" title="b2">O2</font> &amp; not <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">O2</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c3">a</font> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a> <font color="Maroon" title="c2">f</font> holds <br/><font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">O1</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">O2</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c3">a</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">O1</font>, <font color="Maroon" title="c5">O2</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</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="c4">O1</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c5">O2</font> &amp; not <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">O2</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c3">a</font> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a> <font color="Maroon" title="c2">f</font> implies <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">O1</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">O2</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c3">a</font> )</span><br/>



<a NAME="E2:41"/>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">O1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">O1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D1" title="ORDINAL1:def.1">ORDINAL1:def 1</a></span>;<br/>
<a NAME="E3:41"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A2</font></span>: 
<font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O1</font></span>)</span> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">O1</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c3">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="knaster.html#T28" target="_self" title="KNASTER:th.28">Th28</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:41"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<font color="Maroon" title="c4">O1</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c5">O2</font>
 <span class="kw">and </span><br/><a NAME="E5:41"/><span class="lab"><font color="Green" title="E32">A4</font></span>: 
not <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">O2</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c3">a</font> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">and </span><br/><a NAME="E6:41"/><span class="lab"><font color="Green" title="E33">A5</font></span>: 
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">O1</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">O2</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c3">a</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="E7:41"/>
<font color="Maroon" title="c4">O1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">O2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></span>;<br/>
<a NAME="E8:41"/><span class="kw">then </span>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">O2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T33" title="ORDINAL1:th.33">ORDINAL1:33</a></span>;<br/>
<a NAME="E9:41"/><span class="kw">then </span>
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">O2</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O1</font></span>)</span> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c3">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="knaster.html#T28" target="_self" title="KNASTER:th.28">Th28</a></span>;<br/>
<a NAME="E10:41"/><span class="kw">then </span>
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">O1</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O1</font></span>)</span> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c3">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:41"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E3:41"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="lattices.html#T26" title="LATTICES:th.26">LATTICES:26</a></span>;<br/>
<a NAME="E11:41"/><span class="kw">then </span>
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">O1</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">O1</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c3">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="knaster.html#T19" target="_self" title="KNASTER:th.19">Th19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:41"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:41"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E6:41"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="ref" href="abian.html#D4" title="ABIAN:def.4">ABIAN: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/>


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