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<span class="kw">let </span><font color="Maroon" title="c1">G1</font>, <font color="Maroon" title="c2">G2</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</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">f</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b1">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  holds <br/> <a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Olive" title="b1">f</font> is    <a href="lattice4.html#M1" title="LATTICE4:mode.1">Homomorphism</a> of  <a href="group_4.html#K11" title="GROUP_4:func.11">lattice</a> <font color="Maroon" title="c1">G1</font>, <a href="group_4.html#K11" title="GROUP_4:func.11">lattice</a> <font color="Maroon" title="c2">G2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">G2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  implies  <a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Maroon" title="c3">f</font> is    <a href="lattice4.html#M1" title="LATTICE4:mode.1">Homomorphism</a> of  <a href="group_4.html#K11" title="GROUP_4:func.11">lattice</a> <font color="Maroon" title="c1">G1</font>, <a href="group_4.html#K11" title="GROUP_4:func.11">lattice</a> <font color="Maroon" title="c2">G2</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:33"/>
<font color="Maroon" title="c3">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Maroon" title="c3">f</font> is    <a href="lattice4.html#M1" title="LATTICE4:mode.1">Homomorphism</a> of  <a href="group_4.html#K11" title="GROUP_4:func.11">lattice</a> <font color="Maroon" title="c1">G1</font>, <a href="group_4.html#K11" title="GROUP_4:func.11">lattice</a> <font color="Maroon" title="c2">G2</font></span><br/>

<a NAME="E2:33"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A1</font></span>: 
 <a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Maroon" title="c3">f</font> is    <a href="vectsp_8.html#M3" title="VECTSP_8:mode.3">Semilattice-Homomorphism</a> of  <a href="group_4.html#K11" title="GROUP_4:func.11">lattice</a> <font color="Maroon" title="c1">G1</font>, <a href="group_4.html#K11" title="GROUP_4:func.11">lattice</a> <font color="Maroon" title="c2">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="latsubgr.html#T31" target="_self" title="LATSUBGR:th.31">Th36</a></span>;<br/>
<a NAME="E3:33"/>
 <a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Maroon" title="c3">f</font> is    <a href="vectsp_8.html#M4" title="VECTSP_8:mode.4">sup-Semilattice-Homomorphism</a> of  <a href="group_4.html#K11" title="GROUP_4:func.11">lattice</a> <font color="Maroon" title="c1">G1</font>, <a href="group_4.html#K11" title="GROUP_4:func.11">lattice</a> <font color="Maroon" title="c2">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="latsubgr.html#T32" target="_self" title="LATSUBGR:th.32">Th37</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:33"/>
 <a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Maroon" title="c3">f</font> is    <a href="lattice4.html#M1" title="LATTICE4:mode.1">Homomorphism</a> of  <a href="group_4.html#K11" title="GROUP_4:func.11">lattice</a> <font color="Maroon" title="c1">G1</font>, <a href="group_4.html#K11" title="GROUP_4:func.11">lattice</a> <font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="vectsp_8.html#T11" title="VECTSP_8:th.11">VECTSP_8:11</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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