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<span class="kw">let </span><font color="Maroon" title="c1">T</font> be   <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</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">S</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>   <a href="yellow_0.html#V8" title="YELLOW_0:attr.8">sups-inheriting</a>   <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">T</font> holds   <a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> (<font color="Olive" title="b1">S</font>,<font color="Maroon" title="c1">T</font>) is  <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>   <a href="yellow_0.html#V8" title="YELLOW_0:attr.8">sups-inheriting</a>   <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> (<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font>) is  <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</a> </span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">f</font> =  <a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> (<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font>);<br/>
<span class="kw">let </span><font color="Maroon" title="c4">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="waybel_0.html#D33" title="WAYBEL_0:def.33">WAYBEL_0:def 33</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> (<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font>) <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Maroon" title="c4">X</font></span><br/>

<span class="kw">assume </span><a NAME="E1:12"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c2">S</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="waybel_0.html#D31" title="WAYBEL_0:def.31">WAYBEL_0:def 31</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1">(<span class="default"><a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> (<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">T</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> (<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">X</font></span>)</span>,<font color="Maroon" title="c1">T</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> (<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c2">S</font>)</span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E2:12"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1">(<span class="default"><a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> (<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> (<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">X</font></span>)</span>,<font color="Maroon" title="c1">T</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> (<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c2">S</font>)</span>)</span></span><br/>

<a NAME="E3:12"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a></span>;<br/>
<a NAME="E4:12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A1</font></span>: 
 <a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> (<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#D1" title="YELLOW_9:def.1">YELLOW_9:def 1</a></span>;<br/>
<a NAME="E5:12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A2</font></span>: 
<span class="p1">(<span class="default"><a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> (<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T92" title="FUNCT_1:th.92">FUNCT_1:92</a></span>;<br/>
<a NAME="E6:12"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></span>;<br/>
<a NAME="E7:12"/><span class="lab"><font color="Green" title="E10">A4</font></span>: 
<span class="p1">(<span class="default"><a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> (<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font>)</span>)</span> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">X</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> <font color="Maroon" title="c4">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="funct_1.html#T18" title="FUNCT_1:th.18">FUNCT_1:18</a></span>;<br/>
<a NAME="E8:12"/>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">T</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="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="ref" href="yellow_0.html#D19" title="YELLOW_0:def.19">YELLOW_0:def 19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:12"/>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> (<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">X</font></span>)</span>,<font color="Maroon" title="c1">T</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> (<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c2">S</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:12"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E6:12"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E7:12"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="ref" href="yellow_0.html#T64" title="YELLOW_0:th.64">YELLOW_0:64</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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