<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">P</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>   <a href="poset_1.html#V1" title="POSET_1:attr.1">chain-complete</a>  <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</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">g</font> being  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c1">P</font> holds   <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="poset_1.html#K2" title="POSET_1:func.2">iter_min</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">g</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="poset_1.html#K2" title="POSET_1:func.2">iter_min</a> <font color="Olive" title="b1">g</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">g</font> be   <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c1">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="poset_1.html#K2" title="POSET_1:func.2">iter_min</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="poset_1.html#K2" title="POSET_1:func.2">iter_min</a> <font color="Maroon" title="c2">g</font></span>)</span></span>)</span></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c3">L</font> = <font color="Maroon" title="c2">g</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="poset_1.html#K2" title="POSET_1:func.2">iter_min</a> <font color="Maroon" title="c2">g</font></span>)</span> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="orders_2.html#NM2" title="ORDERS_2:NM.2">Chain</a> of <font color="Maroon" title="c1">P</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="poset_1.html#T1" target="_self" title="POSET_1:th.1">ThChain1</a></span>;<br/>
<a NAME="E2:22"/><span class="lab"><font color="Green" title="E18">A00</font></span>: 
(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="poset_1.html#K2" title="POSET_1:func.2">iter_min</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c3">L</font>,<font color="Maroon" title="c1">P</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="poset_1.html#D1" target="_self" title="POSET_1:def.1">DefchainComplete</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">a</font> =  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">P</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">s1</font> =  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="poset_1.html#K2" title="POSET_1:func.2">iter_min</a> <font color="Maroon" title="c2">g</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">s2</font> =  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">L</font>;<br/>
<a NAME="E3:22"/><span class="lab"><font color="Green" title="E19">A201</font></span>: 
 <a href="poset_1.html#K2" title="POSET_1:func.2">iter_min</a> <font color="Maroon" title="c2">g</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="poset_1.html#K2" title="POSET_1:func.2">iter_min</a> <font color="Maroon" title="c2">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E18">A00</font></span></a>, <a class="ref" href="yellow_0.html#D9" title="YELLOW_0:def.9">YELLOW_0:def 9</a></span>;<br/>
<a NAME="E4:22"/><span class="lab"><font color="Green" title="E20">A301</font></span>: 
<font color="Maroon" title="c3">L</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E18">A00</font></span></a>, <a class="ref" href="yellow_0.html#D9" title="YELLOW_0:def.9">YELLOW_0:def 9</a></span>;<br/>
<a NAME="E5:22"/>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="poset_1.html#K2" title="POSET_1:func.2">iter_min</a> <font color="Maroon" title="c2">g</font> holds <br/><font color="Olive" title="b1">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="poset_1.html#K2" title="POSET_1:func.2">iter_min</a> <font color="Maroon" title="c2">g</font> implies <font color="Maroon" title="c7">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">L</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="poset_1.html#K2" title="POSET_1:func.2">iter_min</a> <font color="Maroon" title="c2">g</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="c7">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">L</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font><span class="kw"> such that </span><br/><a NAME="E3:22_1"/><span class="lab"><font color="Green" title="E21">B2</font></span>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> &amp;  ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <font color="Maroon" title="c2">g</font>,<font color="Olive" title="b1">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">P</font></span>)</span> )
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:22_1"/><span class="lab"><font color="Green" title="E22">B3</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c9">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">P</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_1"><span class="lab"><font color="Green" title="E21">B2</font></span></a></span>;<br/>
<a NAME="E6:22_1"/><span class="lab"><font color="Green" title="E23">B4</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> implies <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:22_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</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="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:22_1_1"/><span class="lab"><font color="Green" title="E24">C1</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">z</font> = <span class="p1">(<span class="default"><a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c10">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">P</font></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="poset_1.html#E10"><span class="lab"><font color="Green" title="E9">Lemiter11</font></span></a></span>;<br/>
<a NAME="E5:22_1_1"/>
<font color="Maroon" title="c11">z</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">P</font>
 
;<br/>
<a NAME="E6:22_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">C2</font></span>: 
( <font color="Maroon" title="c11">z</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="c2">g</font> &amp; <font color="Maroon" title="c11">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="poset_1.html#K2" title="POSET_1:func.2">iter_min</a> <font color="Maroon" title="c2">g</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/>
<font color="Maroon" title="c8">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">P</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T73" title="FUNCT_7:th.73">FUNCT_7:73</a>, <a class="txt" href="#E5:22_1"><span class="lab"><font color="Green" title="E22">B3</font></span></a>, <a class="txt" href="#E3:22_1_1"><span class="lab"><font color="Green" title="E24">C1</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="poset_1.html#E11"><span class="lab"><font color="Green" title="E10">Lemiter12</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:22_1_1"/>
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22_1_1"><span class="lab"><font color="Green" title="E25">C2</font></span></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</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="E7:22_1"/>
( <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">P</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_1"><span class="lab"><font color="Green" title="E22">B3</font></span></a>, <a class="txt" href="poset_1.html#E8"><span class="lab"><font color="Green" title="E8">Lemiter10</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:22_1"/>
<font color="Maroon" title="c7">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22_1"><span class="lab"><font color="Green" title="E23">B4</font></span></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="txt" href="#E3:22_1"><span class="lab"><font color="Green" title="E21">B2</font></span></a>, <a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a>, <a class="txt" href="#E4:22"><span class="lab"><font color="Green" title="E20">A301</font></span></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</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="E6:22"/><span class="kw">then </span>
 <a href="poset_1.html#K2" title="POSET_1:func.2">iter_min</a> <font color="Maroon" title="c2">g</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></span>;<br/>
<a NAME="E7:22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A4</font></span>: 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="poset_1.html#K2" title="POSET_1:func.2">iter_min</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E18">A00</font></span></a>, <a class="ref" href="yellow_0.html#T30" title="YELLOW_0:th.30">YELLOW_0:30</a></span>;<br/>
<a NAME="E8:22"/>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L</font> holds <br/><font color="Olive" title="b1">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="poset_1.html#K2" title="POSET_1:func.2">iter_min</a> <font color="Maroon" title="c2">g</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L</font> implies <font color="Maroon" title="c7">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="poset_1.html#K2" title="POSET_1:func.2">iter_min</a> <font color="Maroon" title="c2">g</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">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="c7">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="poset_1.html#K2" title="POSET_1:func.2">iter_min</a> <font color="Maroon" title="c2">g</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:22_2"/><span class="lab"><font color="Green" title="E22">B2</font></span>: 
( <font color="Maroon" title="c8">z</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="c2">g</font> &amp; <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="poset_1.html#K2" title="POSET_1:func.2">iter_min</a> <font color="Maroon" title="c2">g</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">z1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font><span class="kw"> such that </span><br/><a NAME="E5:22_2"/><span class="lab"><font color="Green" title="E23">B3</font></span>: 
( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">z1</font> &amp;  ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c9">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <font color="Maroon" title="c2">g</font>,<font color="Olive" title="b1">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">P</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_2"><span class="lab"><font color="Green" title="E22">B2</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E7:22_2"/><span class="lab"><font color="Green" title="E24">B4</font></span>: 
<font color="Maroon" title="c9">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c10">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">P</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_2"><span class="lab"><font color="Green" title="E23">B3</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">n1</font> = <font color="Maroon" title="c10">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1;<br/>
<font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c10">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">P</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="poset_1.html#E11"><span class="lab"><font color="Green" title="E10">Lemiter12</font></span></a>, <a class="txt" href="#E5:22_2"><span class="lab"><font color="Green" title="E23">B3</font></span></a>, <a class="txt" href="#E7:22_2"><span class="lab"><font color="Green" title="E24">B4</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <font color="Maroon" title="c2">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">P</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T73" title="FUNCT_7:th.73">FUNCT_7:73</a></span>
;<br/>
<a NAME="E9:22_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="poset_1.html#K1" title="POSET_1:func.1">iterSet</a> <font color="Maroon" title="c2">g</font>,<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">P</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_2"><span class="lab"><font color="Green" title="E22">B2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:22_2"/>
<font color="Maroon" title="c7">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="poset_1.html#K2" title="POSET_1:func.2">iter_min</a> <font color="Maroon" title="c2">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22"><span class="lab"><font color="Green" title="E19">A201</font></span></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</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="E9:22"/><span class="kw">then </span>
<font color="Maroon" title="c3">L</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="poset_1.html#K2" title="POSET_1:func.2">iter_min</a> <font color="Maroon" title="c2">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></span>;<br/>
<a NAME="E10:22"/><span class="kw">then </span>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="poset_1.html#K2" title="POSET_1:func.2">iter_min</a> <font color="Maroon" title="c2">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D9" title="YELLOW_0:def.9">YELLOW_0:def 9</a>, <a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E18">A00</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:22"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="poset_1.html#K2" title="POSET_1:func.2">iter_min</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="poset_1.html#K2" title="POSET_1:func.2">iter_min</a> <font color="Maroon" title="c2">g</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="orders_2.html#T25" title="ORDERS_2:th.25">ORDERS_2:25</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>
