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<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</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">w1</font>, <font color="Olive" title="b2">w2</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>   st <font color="Maroon" title="c1">S</font> is <a href="orders_3.html#V1" title="ORDERS_3:attr.1">discrete</a> &amp; <font color="Olive" title="b1">w1</font> <a href="osalg_1.html#R2" title="OSALG_1:pred.2">&lt;=</a> <font color="Olive" title="b2">w2</font> holds <br/><font color="Olive" title="b1">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">w2</font></span><br/><b>let </b><font color="Maroon" title="c2">w1</font>, <font color="Maroon" title="c3">w2</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.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="c1">S</font> is <a href="orders_3.html#V1" title="ORDERS_3:attr.1">discrete</a> &amp; <font color="Maroon" title="c2">w1</font> <a href="osalg_1.html#R2" title="OSALG_1:pred.2">&lt;=</a> <font color="Maroon" title="c3">w2</font> implies <font color="Maroon" title="c2">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w2</font> )</span><br/>





<b>assume </b><a NAME="E1:27"/><i><font color="Green" title="E15">A1</font></i>: 
( <font color="Maroon" title="c1">S</font> is <a href="orders_3.html#V1" title="ORDERS_3:attr.1">discrete</a> &amp; <font color="Maroon" title="c2">w1</font> <a href="osalg_1.html#R2" title="OSALG_1:pred.2">&lt;=</a> <font color="Maroon" title="c3">w2</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">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w2</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c4">S1</font> = <font color="Maroon" title="c1">S</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_3.html#V1" title="ORDERS_3:attr.1">discrete</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a> ;<br/>
<a NAME="E3:27"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">w2</font>
 
<b>by </b><i><a class="txt" href="osalg_1.html#E1:27"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="osalg_1.html#D7" target="_self" title="OSALG_1:def.7">Def7</a></i>;<br/>
<a NAME="E4:27"/><b>then </b><i><font color="Green" title="E16">A2</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">w2</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E5:27"/>
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">w1</font> holds <br/><font color="Maroon" title="c2">w1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">i</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"> ( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">w1</font> implies <font color="Maroon" title="c2">w1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> )</span><br/>

<b>assume </b><a NAME="E1:27_1"/><i><font color="Green" title="E17">A3</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">w1</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">w1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c6">s3</font> = <font color="Maroon" title="c2">w1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>, <font color="Maroon" title="c7">s4</font> = <font color="Maroon" title="c3">w2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="txt" href="osalg_1.html#E4:27"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="osalg_1.html#E1:27_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="partfun1.html#T27" title="PARTFUN1:th.27">PARTFUN1:27</a></i>;<br/>
<a NAME="E3:27_1"/><i><font color="Green" title="E18">A4</font></i>: 
<font color="Maroon" title="c6">s3</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">s4</font>
 
<b>by </b><i><a class="txt" href="osalg_1.html#E1:27"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="osalg_1.html#E1:27_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="osalg_1.html#D7" target="_self" title="OSALG_1:def.7">Def7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">s5</font> = <font color="Maroon" title="c6">s3</font>, <font color="Maroon" title="c9">s6</font> = <font color="Maroon" title="c7">s4</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">S1</font> ;<br/>
<a NAME="E5:27_1"/>
<font color="Maroon" title="c8">s5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">s6</font>
 
<b>by </b><i><a class="txt" href="osalg_1.html#E3:27_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="orders_3.html#T1" title="ORDERS_3:th.1">ORDERS_3:1</a></i>;<br/>
<b>hence </b><a NAME="E6:27_1"/>
<font color="Maroon" title="c2">w1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:27"/>
<font color="Maroon" title="c2">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w2</font>
 <b>by </b><i><a class="txt" href="osalg_1.html#E4:27"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</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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