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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#L1" title="ORDERS_2:struct.1">RelStr</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">R</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">f</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="b3">g</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 <font color="Olive" title="b2">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  holds <br/>( <font color="Olive" title="b3">g</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Olive" title="b1">R</font></span>)</span></span>)</span> is   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Olive" title="b1">R</font></span>)</span></span>)</span>,<font color="Olive" title="b1">R</font> &amp; <font color="Olive" title="b3">g</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Olive" title="b1">R</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Olive" title="b1">R</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">R</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">S</font>; <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="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">g</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 <font color="Olive" title="b1">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  holds <br/>( <font color="Olive" title="b2">g</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span> is   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">R</font> &amp; <font color="Olive" title="b2">g</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  )</span><br/><span class="kw">let </span><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">S</font>,<font color="Maroon" title="c2">T</font>; <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">g</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 <font color="Maroon" title="c4">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  holds <br/>( <font color="Olive" title="b1">g</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span> is   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">R</font> &amp; <font color="Olive" title="b1">g</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">g</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  implies ( <font color="Maroon" title="c5">g</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span> is   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">R</font> &amp; <font color="Maroon" title="c5">g</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  ) )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:22"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
<font color="Maroon" title="c4">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">and </span><br/><a NAME="E2:22"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
 ; <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="c5">g</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span> is   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">R</font> &amp; <font color="Maroon" title="c5">g</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">h</font> = <font color="Maroon" title="c5">g</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span>;<br/>
<a NAME="E3:22"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 
<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="E4:22"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span></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="E5:22"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">R</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="c1">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a></span>;<br/>
<a NAME="E6:22"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span></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="c3">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span></span>)</span> or <font color="Maroon" title="c7">y</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="c3">R</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span></span>)</span>
 ; <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="c7">y</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="c3">R</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:22_1"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:22_1"/><span class="lab"><font color="Green" title="E15">A7</font></span>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c8">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R</font><span class="kw"> such that </span><br/><a NAME="E7:22_1"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#T12" title="YELLOW_2:th.12">YELLOW_2:12</a></span>;<br/>
<a NAME="E8:22_1"/><span class="lab"><font color="Green" title="E17">A9</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="ref" href="waybel10.html#T7" target="_self" title="WAYBEL10:th.7">Th7</a></span>;<br/>
<a NAME="E9:22_1"/><span class="lab"><font color="Green" title="E18">A10</font></span>: 
<font color="Maroon" title="c10">a</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="c3">R</font>
 
;<br/>
<a NAME="E10:22_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="ref" href="waybel10.html#T7" target="_self" title="WAYBEL10:th.7">Th7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:22_1"/>
<font color="Maroon" title="c7">y</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="c3">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E5:22"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E3:22"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E9:22_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E8:22_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</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="E7:22"/>
<font color="Maroon" title="c5">g</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span> is   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></span>; <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="c5">g</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span><br/>

<a NAME="E8:22"/><span class="lab"><font color="Green" title="E14">A11</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">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></span>;<br/>
<a NAME="E9:22"/><span class="lab"><font color="Green" title="E15">A12</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</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="txt" href="#E1:22"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="waybel10.html#T8" target="_self" title="WAYBEL10:th.8">Th8</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E16">A13</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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="c7">x</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 <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> )</span><br/><a NAME="E1:22_2"/><span class="lab"><font color="Green" title="E17">A14</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">R</font>
 <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/><span class="kw">assume </span><a NAME="E2:22_2"/><span class="lab"><font color="Green" title="E18">A15</font></span>: 
<font color="Maroon" title="c7">x</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 <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:22_2"/><span class="lab"><font color="Green" title="E19">A16</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:22_2"/><span class="lab"><font color="Green" title="E20">A17</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22"><span class="lab"><font color="Green" title="E14">A11</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E6:22_2"/><span class="lab"><font color="Green" title="E21">A18</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="txt" href="#E4:22_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E5:22_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></span>;<br/><a NAME="E7:22_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E5:22_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="ref" href="waybel10.html#T7" target="_self" title="WAYBEL10:th.7">Th7</a></span>;<br/><a NAME="E8:22_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E5:22"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E3:22"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E4:22_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E1:22_2"><span class="lab"><font color="Green" title="E17">A14</font></span></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></span>;<br/><span class="kw">hence </span><a NAME="E9:22_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E6:22_2"><span class="lab"><font color="Green" title="E21">A18</font></span></a>, <a class="ref" href="waybel10.html#T7" target="_self" title="WAYBEL10:th.7">Th7</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="E11:22"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:22"/>
<font color="Maroon" title="c5">g</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E8:22"><span class="lab"><font color="Green" title="E14">A11</font></span></a>, <a class="txt" href="#E10:22"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</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/>


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