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<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_6.html#K5" title="FUNCT_6:func.5">&lt;:</a><span class="default"><font color="Olive" title="b1">f</font></span><a href="funct_6.html#K5" title="FUNCT_6:func.5">:&gt;</a></span> holds <br/><font color="Olive" title="b1">f</font> <a href="funct_6.html#K4" title="FUNCT_6:func.4">..</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="funct_6.html#K5" title="FUNCT_6:func.5">&lt;:</a><span class="default"><font color="Olive" title="b1">f</font></span><a href="funct_6.html#K5" title="FUNCT_6:func.5">:&gt;</a></span> <a href="funct_6.html#K4" title="FUNCT_6:func.4">..</a> (<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">f</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="c1">x</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_6.html#K5" title="FUNCT_6:func.5">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="funct_6.html#K5" title="FUNCT_6:func.5">:&gt;</a></span> implies <font color="Maroon" title="c3">f</font> <a href="funct_6.html#K4" title="FUNCT_6:func.4">..</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="funct_6.html#K5" title="FUNCT_6:func.5">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="funct_6.html#K5" title="FUNCT_6:func.5">:&gt;</a></span> <a href="funct_6.html#K4" title="FUNCT_6:func.4">..</a> (<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font>) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:39"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">and </span><br/><a NAME="E2:39"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 <span class="kw">and </span><br/><a NAME="E3:39"/><span class="lab"><font color="Green" title="E21">A2</font></span>: 
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_6.html#K5" title="FUNCT_6:func.5">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="funct_6.html#K5" title="FUNCT_6:func.5">:&gt;</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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> <a href="funct_6.html#K4" title="FUNCT_6:func.4">..</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="funct_6.html#K5" title="FUNCT_6:func.5">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="funct_6.html#K5" title="FUNCT_6:func.5">:&gt;</a></span> <a href="funct_6.html#K4" title="FUNCT_6:func.4">..</a> (<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font>)</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">g</font> = <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c5">h</font> = <span class="p1"><a href="funct_6.html#K5" title="FUNCT_6:func.5">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="funct_6.html#K5" title="FUNCT_6:func.5">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">y</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<a NAME="E5:39"/><span class="lab"><font color="Green" title="E22">A3</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="funct_6.html#T31" target="_self" onmouseover="rs('funct_6/T31')" onmouseout="rh()">Th26</a></span>;<br/>
<a NAME="E6:39"/><span class="lab"><font color="Green" title="E23">A4</font></span>: 
<font color="Maroon" title="c4">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E7:39"/><span class="lab"><font color="Green" title="E24">A5</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c5">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E5:39"><span class="lab"><font color="Green" title="E22">A3</font></span></a></span>;<br/>
<a NAME="E8:39"/>
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="funct_6.html#T32" target="_self" onmouseover="rs('funct_6/T32')" onmouseout="rh()">Th27</a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c3">f</font> <a href="funct_6.html#K4" title="FUNCT_6:func.4">..</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>) = 
<font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="funct_5.html#T38" onmouseover="rs('funct_5/T38')" onmouseout="rh()">FUNCT_5:38</a></span>
<br/>.= 
<font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="funct_6.html#T34" target="_self" onmouseover="rs('funct_6/T34')" onmouseout="rh()">Th29</a></span>
<br/>.= 
<span class="p1"><a href="funct_6.html#K5" title="FUNCT_6:func.5">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="funct_6.html#K5" title="FUNCT_6:func.5">:&gt;</a></span> <a href="funct_6.html#K4" title="FUNCT_6:func.4">..</a> (<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E7:39"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="ref" href="funct_5.html#T38" onmouseover="rs('funct_5/T38')" onmouseout="rh()">FUNCT_5:38</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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