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<b>let </b><font color="Maroon" title="c1">R</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">A</font>, <font color="Olive" title="b2">B</font> being  <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b3">F</font> being   <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>  st <font color="Olive" title="b3">F</font> is <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a> &amp; <font color="Olive" title="b3">F</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> &amp; <font color="Olive" title="b3">F</font> is <a href="osalg_3.html#V1" title="OSALG_3:attr.1">order-sorted</a> holds <br/><font color="Olive" title="b3">F</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a>  is <a href="osalg_3.html#V1" title="OSALG_3:attr.1">order-sorted</a></span><br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">R</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">F</font> being   <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>  st <font color="Olive" title="b1">F</font> is <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a> &amp; <font color="Olive" title="b1">F</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> &amp; <font color="Olive" title="b1">F</font> is <a href="osalg_3.html#V1" title="OSALG_3:attr.1">order-sorted</a> holds <br/><font color="Olive" title="b1">F</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a>  is <a href="osalg_3.html#V1" title="OSALG_3:attr.1">order-sorted</a></span><br/><b>let </b><font color="Maroon" title="c4">F</font> be    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">F</font> is <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a> &amp; <font color="Maroon" title="c4">F</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> &amp; <font color="Maroon" title="c4">F</font> is <a href="osalg_3.html#V1" title="OSALG_3:attr.1">order-sorted</a> implies <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a>  is <a href="osalg_3.html#V1" title="OSALG_3:attr.1">order-sorted</a> )</span><br/>





<b>assume </b><a NAME="E1:8"/><i><font color="Green" title="E7">A1</font></i>: 
( <font color="Maroon" title="c4">F</font> is <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a> &amp; <font color="Maroon" title="c4">F</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> &amp; <font color="Maroon" title="c4">F</font> is <a href="osalg_3.html#V1" title="OSALG_3:attr.1">order-sorted</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a>  is <a href="osalg_3.html#V1" title="OSALG_3:attr.1">order-sorted</a></span><br/>

<b>let </b><font color="Maroon" title="c5">s1</font>, <font color="Maroon" title="c6">s2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="osalg_3.html#D1" target="_self" title="OSALG_3:def.1">OSALG_3:def 1</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">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">s2</font> implies for <font color="Olive" title="b1">a1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s1</font></span>)</span> holds <br/>( <font color="Olive" title="b1">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s2</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a1</font> ) )</span><br/>

<b>assume </b><a NAME="E2:8"/><i><font color="Green" title="E8">A2</font></i>: 
<font color="Maroon" title="c5">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">s2</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">a1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s1</font></span>)</span> holds <br/>( <font color="Olive" title="b1">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s2</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a1</font> )</span><br/>

<a NAME="E3:8"/><i><font color="Green" title="E9">A3</font></i>: 
( <font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s2</font> &amp; <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s2</font> )
 
<b>by </b><i><a class="txt" href="osalg_3.html#E2:8"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="osalg_1.html#D18" title="OSALG_1:def.18">OSALG_1:def 18</a></i>;<br/>
<a NAME="E4:8"/>
( <font color="Maroon" title="c5">s1</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">R</font> &amp; <font color="Maroon" title="c6">s2</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">R</font> )
 
;<br/>
<a NAME="E5:8"/><b>then </b><i><font color="Green" title="E10">A4</font></i>: 
( <font color="Maroon" title="c5">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font> &amp; <font color="Maroon" title="c6">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>let </b><font color="Maroon" title="c7">a1</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="c7">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s1</font></span>)</span> implies ( <font color="Maroon" title="c7">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s2</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a1</font> ) )</span><br/>

<b>assume </b><a NAME="E6:8"/><i><font color="Green" title="E11">A5</font></i>: 
<font color="Maroon" title="c7">a1</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c5">s1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s2</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a1</font> )</span><br/>

<a NAME="E7:8"/><i><font color="Green" title="E12">A6</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c5">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s1</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s2</font> )
 
<b>by </b><i><a class="txt" href="osalg_3.html#E1:8"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="msualg_3.html#D3" title="MSUALG_3:def.3">MSUALG_3:def 3</a></i>;<br/>
<a NAME="E8:8"/><i><font color="Green" title="E13">A7</font></i>: 
( <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c5">s1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 
<b>by </b><i><a class="txt" href="osalg_3.html#E1:8"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="osalg_3.html#E5:8"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="msualg_3.html#D2" title="MSUALG_3:def.2">MSUALG_3:def 2</a></i>;<br/>
<a NAME="E9:8"/><b>then </b><i><font color="Green" title="E14">A8</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c5">s1</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s2</font> )
 
<b>by </b><i><a class="txt" href="osalg_3.html#E7:8"><i><font color="Green" title="E12">A6</font></i></a>, <a class="ref" href="funct_2.html#T31" title="FUNCT_2:th.31">FUNCT_2:31</a></i>;<br/>
<a NAME="E10:8"/><i><font color="Green" title="E15">A9</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c5">s1</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  )
 
<b>by </b><i><a class="txt" href="osalg_3.html#E1:8"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="msualg_3.html#D5" title="MSUALG_3:def.5">MSUALG_3:def 5</a></i>;<br/>
<a NAME="E11:8"/><i><font color="Green" title="E16">A10</font></i>: 
<font color="Maroon" title="c7">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s1</font>
 
<b>by </b><i><a class="txt" href="osalg_3.html#E6:8"><i><font color="Green" title="E11">A5</font></i></a></i>;<br/>
<a NAME="E12:8"/><b>then </b><i><font color="Green" title="E17">A11</font></i>: 
( <font color="Maroon" title="c7">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s2</font> &amp; <font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="osalg_3.html#E3:8"><i><font color="Green" title="E9">A3</font></i></a></i>;<br/>
<a NAME="E13:8"/><i><font color="Green" title="E18">A12</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c5">s1</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c5">s1</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="osalg_3.html#E6:8"><i><font color="Green" title="E11">A5</font></i></a>, <a class="txt" href="osalg_3.html#E10:8"><i><font color="Green" title="E15">A9</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E14:8"/><i><font color="Green" title="E19">A13</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c5">s1</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s1</font>
 
<b>by </b><i><a class="txt" href="osalg_3.html#E9:8"><i><font color="Green" title="E14">A8</font></i></a>, <a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E15:8"/><b>then </b><i><font color="Green" title="E20">A14</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c5">s1</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s1</font> &amp; <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="osalg_3.html#E13:8"><i><font color="Green" title="E18">A12</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:8"/>
<font color="Maroon" title="c7">a1</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s2</font></span>)</span>
 <b>by </b><i><a class="txt" href="osalg_3.html#E3:8"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="osalg_3.html#E12:8"><i><font color="Green" title="E17">A11</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a1</font></span><br/>

<a NAME="E17:8"/><i><font color="Green" title="E21">A15</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c5">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s1</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s2</font> )
 
<b>by </b><i><a class="txt" href="osalg_3.html#E3:8"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="osalg_3.html#E11:8"><i><font color="Green" title="E16">A10</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">c1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c5">s1</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a1</font>;<br/>
<b>set </b><font color="Maroon" title="c9">c2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a1</font>;<br/>
<a NAME="E18:8"/><i><font color="Green" title="E22">A16</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a1</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s2</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c5">s1</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a1</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="osalg_3.html#E3:8"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="osalg_3.html#E9:8"><i><font color="Green" title="E14">A8</font></i></a>, <a class="txt" href="osalg_3.html#E11:8"><i><font color="Green" title="E16">A10</font></i></a>, <a class="txt" href="osalg_3.html#E15:8"><i><font color="Green" title="E20">A14</font></i></a>, <a class="txt" href="osalg_3.html#E17:8"><i><font color="Green" title="E21">A15</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a1</font></span>)</span> = 
<font color="Maroon" title="c7">a1</font>
<b>by </b><i><a class="txt" href="osalg_3.html#E3:8"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="osalg_3.html#E7:8"><i><font color="Green" title="E12">A6</font></i></a>, <a class="txt" href="osalg_3.html#E8:8"><i><font color="Green" title="E13">A7</font></i></a>, <a class="txt" href="osalg_3.html#E11:8"><i><font color="Green" title="E16">A10</font></i></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c5">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c5">s1</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a1</font></span>)</span>
<b>by </b><i><a class="txt" href="osalg_3.html#E6:8"><i><font color="Green" title="E11">A5</font></i></a>, <a class="txt" href="osalg_3.html#E7:8"><i><font color="Green" title="E12">A6</font></i></a>, <a class="txt" href="osalg_3.html#E8:8"><i><font color="Green" title="E13">A7</font></i></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c5">s1</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a1</font></span>)</span>
<b>by </b><i><a class="txt" href="osalg_3.html#E1:8"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="osalg_3.html#E2:8"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="osalg_3.html#E13:8"><i><font color="Green" title="E18">A12</font></i></a>, <a class="txt" href="osalg_3.html#E14:8"><i><font color="Green" title="E19">A13</font></i></a>, <a class="txt" href="osalg_3.html#E17:8"><i><font color="Green" title="E21">A15</font></i></a>, <a class="ref" href="osalg_3.html#D1" target="_self" title="OSALG_3:def.1">Def1</a></i>
;<br/>
<b>hence </b><a NAME="E20:8"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a1</font>
 <b>by </b><i><a class="txt" href="osalg_3.html#E8:8"><i><font color="Green" title="E13">A7</font></i></a>, <a class="txt" href="osalg_3.html#E10:8"><i><font color="Green" title="E15">A9</font></i></a>, <a class="txt" href="osalg_3.html#E18:8"><i><font color="Green" title="E22">A16</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
