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<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">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="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>  st  ex <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c2">x</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a>  <a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">n</font> &amp; ( for <font color="Olive" title="b3">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">y</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a>  <a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">n</font> holds <br/><font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y</font> ) ) holds <br/><font color="Maroon" title="c2">x</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Olive" title="b1">f</font></span><br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<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"> (  ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c2">x</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a>  <a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c3">f</font>,<font color="Olive" title="b1">n</font> &amp; ( for <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">y</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a>  <a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c3">f</font>,<font color="Olive" title="b1">n</font> holds <br/><font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> ) ) implies <font color="Maroon" title="c2">x</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c3">f</font> )</span><br/>





<b>given </b><font color="Maroon" title="c4">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><a NAME="E2:3"/><i><font color="Green" title="E2">A1</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a>  <a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">n</font>
 <b>and </b><br/><a NAME="E3:3"/><i><font color="Green" title="E3">A2</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a>  <a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">n</font> holds <br/><font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c3">f</font></span><br/>

<a NAME="E4:3"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#T67" title="FUNCT_2:th.67">FUNCT_2:67</a></i>;<br/>
<b>hence </b><a NAME="E5:3"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="knaster.html#E2:3"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="abian.html#D3" title="ABIAN:def.3">ABIAN:def 3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="abian.html#D3" title="ABIAN:def.3">ABIAN:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span><br/>

<b>thus </b><a NAME="E6:3"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="knaster.html#E2:3"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="knaster.html#E3:3"><i><font color="Green" title="E3">A2</font></i></a>, <a class="ref" href="knaster.html#T4" target="_self" title="KNASTER:th.4">Th4</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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