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<b>let </b><font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c2">L2</font>, <font color="Maroon" title="c3">L3</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="orders_2.html#V3" title="ORDERS_2:attr.3">transitive</a> <a href="orders_2.html#V4" title="ORDERS_2:attr.4">antisymmetric</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:23"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c4">f</font> is <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a>
 <b>and </b><br/><a NAME="E2:23"/><i><font color="Green" title="E20">A2</font></i>: 
<font color="Maroon" title="c2">L2</font> is  <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a> <a href="yellow_0.html#V7" title="YELLOW_0:attr.7">infs-inheriting</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c3">L3</font>
 <b>and </b><br/><a NAME="E3:23"/><i><font color="Green" title="E21">A3</font></i>: 
<font color="Maroon" title="c3">L3</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>
 ;<br/>

<a NAME="E4:23"/>
( not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</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="c3">L3</font> )
 
<b>by </b><i><a class="txt" href="waybel13.html#E2:23"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">g</font> = <font color="Maroon" title="c4">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c3">L3</font> <b>by </b><i><a class="ref" href="funct_2.html#T9" title="FUNCT_2:th.9">FUNCT_2:9</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">g</font>
;<br/>

<b>thus </b><a NAME="E6:23"/>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">g</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L1</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1"/><i><font color="Green" title="E22">A4</font></i>: 
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c1">L1</font>
 ;<br/><a NAME="E2:23_1_1"/><i><font color="Green" title="E23">A5</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="waybel_0.html#R3" title="WAYBEL_0:pred.3">preserves_inf_of</a> <font color="Maroon" title="c6">X</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E1:23"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="waybel_0.html#D32" title="WAYBEL_0:def.32">WAYBEL_0:def 32</a></i>;<br/><b>thus </b><a NAME="E3:23_1_1"/><i><font color="Green" title="E24">A6</font></i>: 
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c5">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c3">L3</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E3:23"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></i>;<br/><a NAME="E4:23_1_1"/><b>then </b>
 <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">X</font></span>)</span>,<font color="Maroon" title="c3">L3</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">L2</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E2:23"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="yellow_0.html#D18" title="YELLOW_0:def.18">YELLOW_0:def 18</a></i>;<br/><b>hence </b> <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">X</font></span>)</span> = 
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">X</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel13.html#E2:23"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="waybel13.html#E3:23_1_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="yellow_0.html#T64" title="YELLOW_0:th.64">YELLOW_0:64</a></i>
<br/>.= 
<font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c6">X</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel13.html#E1:23_1_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="waybel13.html#E2:23_1_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="waybel_0.html#D30" title="WAYBEL_0:def.30">WAYBEL_0:def 30</a></i>
;<br/><br/></div><b>end;</b></div><b>hence </b><a NAME="E2:23_1"/>
<font color="Maroon" title="c5">g</font> <a href="waybel_0.html#R3" title="WAYBEL_0:pred.3">preserves_inf_of</a> <font color="Maroon" title="c6">X</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D30" title="WAYBEL_0:def.30">WAYBEL_0:def 30</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:23"/>
<font color="Maroon" title="c5">g</font> is <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#D32" title="WAYBEL_0:def.32">WAYBEL_0:def 32</a></i>;<br/>


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