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<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being <a href="cohsp_1.html#V6" title="COHSP_1:attr.6">V191</a> <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font>, <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">S</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">S</font> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a> <font color="Olive" title="b1">f</font> holds <br/>(  <a href="knaster.html#K1" title="KNASTER:func.1">lfp</a> <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">S</font> &amp; <font color="Olive" title="b2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="knaster.html#K2" title="KNASTER:func.2">gfp</a> <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">f</font> )</span><br/><b>let </b><font color="Maroon" title="c2">f</font> be  <a href="cohsp_1.html#V6" title="COHSP_1:attr.6">V191</a> <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font>, <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">S</font> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a> <font color="Maroon" title="c2">f</font> holds <br/>(  <a href="knaster.html#K1" title="KNASTER:func.1">lfp</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="knaster.html#K2" title="KNASTER:func.2">gfp</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font> )</span><br/><b>let </b><font color="Maroon" title="c3">S</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">S</font> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a> <font color="Maroon" title="c2">f</font> implies (  <a href="knaster.html#K1" title="KNASTER:func.1">lfp</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">S</font> &amp; <font color="Maroon" title="c3">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="knaster.html#K2" title="KNASTER:func.2">gfp</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font> ) )</span><br/>





<b>assume </b><a NAME="E1:11"/>
<font color="Maroon" title="c3">S</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> <font color="Maroon" title="c3">S</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="abian.html#D4" title="ABIAN:def.4">ABIAN:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="knaster.html#K1" title="KNASTER:func.1">lfp</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">S</font> &amp; <font color="Maroon" title="c3">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="knaster.html#K2" title="KNASTER:func.2">gfp</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font> )</span><br/>

<b>hence </b><a NAME="E2:11"/>
(  <a href="knaster.html#K1" title="KNASTER:func.1">lfp</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">S</font> &amp; <font color="Maroon" title="c3">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="knaster.html#K2" title="KNASTER:func.2">gfp</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font> )
 <b>by </b><i><a class="ref" href="knaster.html#T8" target="_self" title="KNASTER:th.8">Th8</a>, <a class="ref" href="knaster.html#T9" target="_self" title="KNASTER:th.9">Th9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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