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<span class="kw">let </span><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P</font> being   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><br/>  for <font color="Olive" title="b2">p3</font>, <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">p1</font>, <font color="Olive" title="b5">p4</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">P</font><br/>  for <font color="Olive" title="b6">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="Olive" title="b1">P</font>,<font color="Olive" title="b1">P</font>  st <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font> &amp; <font color="Olive" title="b2">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">p</font> &amp; <font color="Olive" title="b3">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">p3</font> &amp; <font color="Olive" title="b4">p1</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="Olive" title="b6">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">p</font> &amp; <font color="Olive" title="b5">p4</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="Olive" title="b6">g</font>,<font color="Maroon" title="c2">m</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">p</font> holds <br/><font color="Olive" title="b4">p1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b5">p4</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p3</font>, <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">p1</font>, <font color="Olive" title="b4">p4</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">P</font><br/>  for <font color="Olive" title="b5">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="c3">P</font>,<font color="Maroon" title="c3">P</font>  st <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font> &amp; <font color="Olive" title="b1">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font> &amp; <font color="Olive" title="b2">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">p3</font> &amp; <font color="Olive" title="b3">p1</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="Olive" title="b5">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">p</font> &amp; <font color="Olive" title="b4">p4</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="Olive" title="b5">g</font>,<font color="Maroon" title="c2">m</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">p</font> holds <br/><font color="Olive" title="b3">p1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b4">p4</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">p3</font>, <font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c6">p1</font>, <font color="Maroon" title="c7">p4</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c3">P</font>,<font color="Maroon" title="c3">P</font>  st <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font> &amp; <font color="Maroon" title="c4">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> &amp; <font color="Maroon" title="c5">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">p3</font> &amp; <font color="Maroon" title="c6">p1</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="Olive" title="b1">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">p</font> &amp; <font color="Maroon" title="c7">p4</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="Olive" title="b1">g</font>,<font color="Maroon" title="c2">m</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">p</font> holds <br/><font color="Maroon" title="c6">p1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">p4</font></span><br/><span class="kw">let </span><font color="Maroon" title="c8">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="c3">P</font>,<font color="Maroon" title="c3">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font> &amp; <font color="Maroon" title="c4">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> &amp; <font color="Maroon" title="c5">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">p3</font> &amp; <font color="Maroon" title="c6">p1</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="c8">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">p</font> &amp; <font color="Maroon" title="c7">p4</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="c8">g</font>,<font color="Maroon" title="c2">m</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">p</font> implies <font color="Maroon" title="c6">p1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">p4</font> )</span><br/>















<span class="kw">assume </span><a NAME="E1:17"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font> &amp; <font color="Maroon" title="c4">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> &amp; <font color="Maroon" title="c5">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">p3</font> &amp; <font color="Maroon" title="c6">p1</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="c8">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">p</font> &amp; <font color="Maroon" title="c7">p4</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="c8">g</font>,<font color="Maroon" title="c2">m</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">p</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">p1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">p4</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">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="E3:17"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">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">thus </span><a NAME="E4:17"/>
<font color="Maroon" title="c6">p1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">p4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="poset_1.html#E13"><span class="lab"><font color="Green" title="E12">Lm9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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