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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">T</font> being   <a href="abcmiz_1.html#M3" title="ABCMIZ_1:mode.3">quasi-type</a> of <font color="Maroon" title="c1">C</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">T</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K48" title="ABCMIZ_1:func.48">vars</a> <font color="Olive" title="b1">T</font> );<br/>
<a NAME="E1:206_1_2"/><i><font color="Green" title="E100">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_1.html#K39" title="ABCMIZ_1:func.39">QuasiTypes</a> <font color="Maroon" title="c1">C</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="206_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_1.html#K39" title="ABCMIZ_1:func.39">QuasiTypes</a> <font color="Maroon" title="c1">C</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<b>assume </b><a NAME="E1:206_1_2_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_1.html#K39" title="ABCMIZ_1:func.39">QuasiTypes</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<b>then reconsider </b><font color="Maroon" title="c3">T</font> = <font color="Maroon" title="c2">x</font> as    <a href="abcmiz_1.html#M3" title="ABCMIZ_1:mode.3">quasi-type</a> of <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="ref" href="abcmiz_1.html#D43" target="_self" title="ABCMIZ_1:def.43">QUASITYPE</a></i>;<br/>
<b>take </b><font color="Maroon" title="c4">y</font> =  <a href="abcmiz_1.html#K48" title="ABCMIZ_1:func.48">vars</a> <font color="Maroon" title="c3">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font>]</span><br/>

<b>take </b>
<font color="Maroon" title="c3">T</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K48" title="ABCMIZ_1:func.48">vars</a> <font color="Maroon" title="c3">T</font> )</span><br/>

<b>thus </b><a NAME="E3:206_1_2_1"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K48" title="ABCMIZ_1:func.48">vars</a> <font color="Maroon" title="c3">T</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c2">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:206_1_2"/><i><font color="Green" title="E101">A3</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K39" title="ABCMIZ_1:func.39">QuasiTypes</a> <font color="Maroon" title="c1">C</font>
 <b>and </b><br/><a NAME="E4:206_1_2"/><i><font color="Green" title="E102">A4</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_1.html#K39" title="ABCMIZ_1:func.39">QuasiTypes</a> <font color="Maroon" title="c1">C</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="classes1.html#S1" title="CLASSES1:sch.1">CLASSES1:sch 1</a>(<a class="txt" href="#E1:206_1_2"><i><font color="Green" title="E100">A1</font></i></a>);<br/></i>
<a NAME="E5:206_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</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 <a href="abcmiz_1.html#K49" title="ABCMIZ_1:func.49">VarPoset</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="206_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> or <font color="Maroon" title="c3">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 <a href="abcmiz_1.html#K49" title="ABCMIZ_1:func.49">VarPoset</a>  )</span><br/>

<b>assume </b><a NAME="E1:206_1_2_2"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">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 <a href="abcmiz_1.html#K49" title="ABCMIZ_1:func.49">VarPoset</a> </span><br/>

<b>then consider </b><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:206_1_2_2"/><i><font color="Green" title="E103">A5</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">T</font> being    <a href="abcmiz_1.html#M3" title="ABCMIZ_1:mode.3">quasi-type</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E5:206_1_2_2"/><i><font color="Green" title="E104">A6</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">T</font> &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K48" title="ABCMIZ_1:func.48">vars</a> <font color="Maroon" title="c5">T</font> )
 <b>by </b><i><a class="txt" href="#E3:206_1_2"><i><font color="Green" title="E101">A3</font></i></a>, <a class="txt" href="#E4:206_1_2"><i><font color="Green" title="E102">A4</font></i></a>, <a class="txt" href="#E3:206_1_2_2"><i><font color="Green" title="E103">A5</font></i></a></i>;<br/>
<a NAME="E6:206_1_2_2"/>
 <a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K48" title="ABCMIZ_1:func.48">vars</a> <font color="Maroon" title="c5">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K48" title="ABCMIZ_1:func.48">vars</a> <font color="Maroon" title="c5">T</font>
 
;<br/>
<a NAME="E7:206_1_2_2"/><b>then </b>
<font color="Maroon" title="c3">y</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="abcmiz_1.html#K49" title="ABCMIZ_1:func.49">VarPoset</a> 
 
<b>by </b><i><a class="txt" href="#E5:206_1_2_2"><i><font color="Green" title="E104">A6</font></i></a>, <a class="ref" href="abcmiz_1.html#T110" target="_self" title="ABCMIZ_1:th.110">Th21</a></i>;<br/>
<b>hence </b><a NAME="E8:206_1_2_2"/>
<font color="Maroon" title="c3">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 <a href="abcmiz_1.html#K49" title="ABCMIZ_1:func.49">VarPoset</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c3">f</font> = <font color="Maroon" title="c2">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="abcmiz_1.html#K39" title="ABCMIZ_1:func.39">QuasiTypes</a> <font color="Maroon" title="c1">C</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="abcmiz_1.html#K49" title="ABCMIZ_1:func.49">VarPoset</a>  <b>by </b><i><a class="txt" href="#E3:206_1_2"><i><font color="Green" title="E101">A3</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">f</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">T</font> being   <a href="abcmiz_1.html#M3" title="ABCMIZ_1:mode.3">quasi-type</a> of <font color="Maroon" title="c1">C</font> holds  <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K48" title="ABCMIZ_1:func.48">vars</a> <font color="Olive" title="b1">T</font></span><br/>

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="abcmiz_1.html#M3" title="ABCMIZ_1:mode.3">quasi-type</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K48" title="ABCMIZ_1:func.48">vars</a> <font color="Maroon" title="c4">x</font></span><br/>

<a NAME="E7:206_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_1.html#K39" title="ABCMIZ_1:func.39">QuasiTypes</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#D43" target="_self" title="ABCMIZ_1:def.43">QUASITYPE</a></i>;<br/>
<a NAME="E8:206_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">T</font> being   <a href="abcmiz_1.html#M3" title="ABCMIZ_1:mode.3">quasi-type</a> of <font color="Maroon" title="c1">C</font> st <br/>( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">T</font> &amp; <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K48" title="ABCMIZ_1:func.48">vars</a> <font color="Olive" title="b1">T</font> )
 
<b>by </b><i><a class="txt" href="#E4:206_1_2"><i><font color="Green" title="E102">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:206_1_2"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K48" title="ABCMIZ_1:func.48">vars</a> <font color="Maroon" title="c4">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
