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<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">Y</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Z</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">Z</font> <a href="yellow_1.html#K6">|^</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/><b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <span class="p1">(<span class="default"><font color="Maroon">Z</font> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/>









<b>assume </b><a NAME="E1:1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">f</font> is <a href="waybel27.html#V2">currying</a> &amp; <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">f</font> is <a href="funct_2.html#V2">onto</a> )
 ;<br/>

<a NAME="E2:1"/><b>then </b>
( <font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="funct_2.html#V2">onto</a> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:1"/><b>then </b><i><font color="Green">E47</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> )
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E4:1"/><i><font color="Green">E48</font></i>: 
(  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Z</font> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Z</font> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">X</font></span>)</span> &amp;  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">Y</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Z</font> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">Y</font></span>)</span> &amp;  <a href="funct_2.html#K1">Funcs</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Z</font> <a href="yellow_1.html#K6">|^</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> )
 
<b>by </b><i><a class="ref" href="yellow_1.html#T28">YELLOW_1:28</a></i>;<br/>
<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="waybel27.html#D1">WAYBEL27:def 1</a>
<div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span>
 ;<br/><a NAME="E2:1_1"/><b>then </b>
<font color="Maroon">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Z</font> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T59">YELLOW_0:59</a></i>;<br/><a NAME="E3:1_1"/><b>then </b>
<font color="Maroon">x</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>, <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">Y</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/><b>hence </b><a NAME="E4:1_1"/>
<font color="Maroon">x</font> is  <a href="funcop_1.html#V1">Function-yielding</a> <a href="funct_1.html#NM1">Function</a>
 ;<br/></div>
<b>end;</b></div>

<a NAME="E6:1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> 
 
<b>by </b><i>, , <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>;<br/>
<b>let </b><font color="Maroon">g</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><a NAME="E7:1"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">h</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">h</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i>, , <a class="ref" href="waybel27.html#D2">WAYBEL27:def 2</a></i>;<br/>
<a NAME="E11:1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="funct_5.html#K3">curry</a> <font color="Maroon">h</font>
 
<b>by </b><i>, , <a class="ref" href="waybel27.html#D2">WAYBEL27:def 2</a></i>;<br/>
<a NAME="E12:1"/>
<font color="Maroon">h</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">Z</font> <a href="yellow_1.html#K6">|^</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T59">YELLOW_0:59</a></i>;<br/>
<a NAME="E13:1"/><b>then </b>
<font color="Maroon">h</font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<a NAME="E14:1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E15:1"/><b>then </b>
 <a href="funct_5.html#K4">uncurry</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 
<b>by </b><i>, <a class="ref" href="funct_5.html#T56">FUNCT_5:56</a></i>;<br/>
<b>hence </b><a NAME="E16:1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="funct_5.html#K4">uncurry</a> <font color="Maroon">g</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>;<br/>


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