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<span class="kw">let </span><font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c2">L2</font> be   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font> holds <br/> ( <font color="Olive" title="b1">f</font> is    <a href="lattice4.html#M1" title="LATTICE4:mode.1">Homomorphism</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font> iff ( <font color="Olive" title="b1">f</font> is    <a href="vectsp_8.html#M4" title="VECTSP_8:mode.4">sup-Semilattice-Homomorphism</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font> &amp; <font color="Olive" title="b1">f</font> is    <a href="vectsp_8.html#M3" title="VECTSP_8:mode.3">Semilattice-Homomorphism</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> is    <a href="lattice4.html#M1" title="LATTICE4:mode.1">Homomorphism</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font> iff ( <font color="Maroon" title="c3">f</font> is    <a href="vectsp_8.html#M4" title="VECTSP_8:mode.4">sup-Semilattice-Homomorphism</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font> &amp; <font color="Maroon" title="c3">f</font> is    <a href="vectsp_8.html#M3" title="VECTSP_8:mode.3">Semilattice-Homomorphism</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font> ) )</span><br/>



<a NAME="E1:19"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
( <font color="Maroon" title="c3">f</font> is    <a href="vectsp_8.html#M4" title="VECTSP_8:mode.4">sup-Semilattice-Homomorphism</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font> &amp; <font color="Maroon" title="c3">f</font> is    <a href="vectsp_8.html#M3" title="VECTSP_8:mode.3">Semilattice-Homomorphism</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font> implies <font color="Maroon" title="c3">f</font> is    <a href="lattice4.html#M1" title="LATTICE4:mode.1">Homomorphism</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:19_1"/>
( <font color="Maroon" title="c3">f</font> is    <a href="vectsp_8.html#M4" title="VECTSP_8:mode.4">sup-Semilattice-Homomorphism</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font> &amp; <font color="Maroon" title="c3">f</font> is    <a href="vectsp_8.html#M3" title="VECTSP_8:mode.3">Semilattice-Homomorphism</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</font> is    <a href="lattice4.html#M1" title="LATTICE4:mode.1">Homomorphism</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><br/>

<a NAME="E2:19_1"/><span class="kw">then </span>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font> holds <br/> ( <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">b</font></span>)</span> &amp; <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">b</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_8.html#D8" target="_self" title="VECTSP_8:def.8">Def8</a>, <a class="ref" href="vectsp_8.html#D9" target="_self" title="VECTSP_8:def.9">Def9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:19_1"/>
<font color="Maroon" title="c3">f</font> is    <a href="lattice4.html#M1" title="LATTICE4:mode.1">Homomorphism</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice4.html#D1" title="LATTICE4:def.1">LATTICE4:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:19"/>
( <font color="Maroon" title="c3">f</font> is    <a href="lattice4.html#M1" title="LATTICE4:mode.1">Homomorphism</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font> implies ( <font color="Maroon" title="c3">f</font> is    <a href="vectsp_8.html#M4" title="VECTSP_8:mode.4">sup-Semilattice-Homomorphism</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font> &amp; <font color="Maroon" title="c3">f</font> is    <a href="vectsp_8.html#M3" title="VECTSP_8:mode.3">Semilattice-Homomorphism</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:19_2"/>
<font color="Maroon" title="c3">f</font> is    <a href="lattice4.html#M1" title="LATTICE4:mode.1">Homomorphism</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> is    <a href="vectsp_8.html#M4" title="VECTSP_8:mode.4">sup-Semilattice-Homomorphism</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font> &amp; <font color="Maroon" title="c3">f</font> is    <a href="vectsp_8.html#M3" title="VECTSP_8:mode.3">Semilattice-Homomorphism</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font> )</span><br/>

<a NAME="E2:19_2"/><span class="kw">then </span>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font> holds <br/> ( <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">b</font></span>)</span> &amp; <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">b</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice4.html#D1" title="LATTICE4:def.1">LATTICE4:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:19_2"/>
( <font color="Maroon" title="c3">f</font> is    <a href="vectsp_8.html#M4" title="VECTSP_8:mode.4">sup-Semilattice-Homomorphism</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font> &amp; <font color="Maroon" title="c3">f</font> is    <a href="vectsp_8.html#M3" title="VECTSP_8:mode.3">Semilattice-Homomorphism</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_8.html#D8" target="_self" title="VECTSP_8:def.8">Def8</a>, <a class="ref" href="vectsp_8.html#D9" target="_self" title="VECTSP_8:def.9">Def9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:19"/>
( <font color="Maroon" title="c3">f</font> is    <a href="lattice4.html#M1" title="LATTICE4:mode.1">Homomorphism</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font> iff ( <font color="Maroon" title="c3">f</font> is    <a href="vectsp_8.html#M4" title="VECTSP_8:mode.4">sup-Semilattice-Homomorphism</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font> &amp; <font color="Maroon" title="c3">f</font> is    <a href="vectsp_8.html#M3" title="VECTSP_8:mode.3">Semilattice-Homomorphism</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E13">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
