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<span class="kw">let </span><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">T</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font><br/>  for <font color="Olive" title="b2">d</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c1">S</font>  st <span class="p1"><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">[</a><span class="default"><font color="Olive" title="b1">g</font>,<font color="Olive" title="b2">d</font></span><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">]</a></span> is  <a href="waybel_1.html#V3" title="WAYBEL_1:attr.3">Galois</a>  holds <br/>( <font color="Olive" title="b1">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  iff <font color="Olive" title="b2">d</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">g</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>; <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">d</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c1">S</font>  st <span class="p1"><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">g</font>,<font color="Olive" title="b1">d</font></span><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">]</a></span> is  <a href="waybel_1.html#V3" title="WAYBEL_1:attr.3">Galois</a>  holds <br/>( <font color="Maroon" title="c3">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  iff <font color="Olive" title="b1">d</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">d</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">d</font></span><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">]</a></span> is  <a href="waybel_1.html#V3" title="WAYBEL_1:attr.3">Galois</a>  implies ( <font color="Maroon" title="c3">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  iff <font color="Maroon" title="c4">d</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:43"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
<span class="p1"><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">d</font></span><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">]</a></span> is  <a href="waybel_1.html#V3" title="WAYBEL_1:attr.3">Galois</a> 
 ; <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">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  iff <font color="Maroon" title="c4">d</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  )</span><br/>

<div><span class="kw">hereby </span> <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="c4">d</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  implies <font color="Maroon" title="c3">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )</span>
<div class="add"><a NAME="E1:43_1"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
( <font color="Maroon" title="c4">d</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">T</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="waybel_1.html#T18" target="_self" title="WAYBEL_1:th.18">Th19</a></span>;<br/><a NAME="E2:43_1"/>
( <font color="Maroon" title="c3">g</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>  &amp; <font color="Maroon" title="c4">d</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="waybel_1.html#T8" target="_self" title="WAYBEL_1:th.8">Th9</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A3</font></span>: <font color="Maroon" title="c3">g</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="ref" href="waybel_1.html#T20" target="_self" title="WAYBEL_1:th.20">Th21</a></span>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T36" title="RELAT_1:th.36">RELAT_1:36</a></span>
;<br/><a NAME="E4:43_1"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font> &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E5:43_1"/><span class="lab"><font color="Green" title="E37">A5</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <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">S</font>
 ;<br/><span class="kw">assume </span><a NAME="E6:43_1"/>
<font color="Maroon" title="c3">g</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"> <font color="Maroon" title="c4">d</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> </span><br/><a NAME="E7:43_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">d</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_1"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="txt" href="#E5:43_1"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E3:43_1"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="funct_1.html#T28" title="FUNCT_1:th.28">FUNCT_1:28</a></span>;<br/><span class="kw">hence </span><a NAME="E8:43_1"/>
<font color="Maroon" title="c4">d</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</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/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E3:43"/>
<font color="Maroon" title="c4">d</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 ; <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">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> </span><br/>

<a NAME="E4:43"/><span class="kw">then </span>
 for <font color="Olive" title="b1">s</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> <a href="waybel_1.html#R4" title="WAYBEL_1:pred.4">is_maximum_of</a> <font color="Maroon" title="c4">d</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">s</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="waybel_1.html#T25" target="_self" title="WAYBEL_1:th.25">Th27</a></span>;<br/>
<a NAME="E5:43"/><span class="kw">then </span>
<font color="Maroon" title="c4">d</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T26" target="_self" title="WAYBEL_1:th.26">Th28</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:43"/>
<font color="Maroon" title="c3">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</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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