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<span class="kw">set </span><font color="Maroon" title="c1">ad</font> = <a href="binop_2.html#K9" title="BINOP_2:func.9">addreal</a> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>;<br/>
<a NAME="E1:54"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>,<a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T12" onmouseover="rs('numbers/T12')" onmouseout="rh()">NUMBERS:12</a>, <a class="ref" href="zfmisc_1.html#T96" onmouseover="rs('zfmisc_1/T96')" onmouseout="rh()">ZFMISC_1:96</a></span>;<br/>
<a NAME="E2:54"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>,<a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <a href="binop_2.html#K9" title="BINOP_2:func.9">addreal</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E3:54"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A2</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K9" title="BINOP_2:func.9">addreal</a> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>,<a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T62" onmouseover="rs('relat_1/T62')" onmouseout="rh()">RELAT_1:62</a></span>;<br/>
<a NAME="E4:54"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <a href="binop_2.html#K15" title="BINOP_2:func.15">addrat</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>,<a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E5:54"/>
 for <font color="Olive" title="b1">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">z</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">(<span class="default"><a href="binop_2.html#K9" title="BINOP_2:func.9">addreal</a> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="binop_2.html#K9" title="BINOP_2:func.9">addreal</a> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K15" title="BINOP_2:func.15">addrat</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="54_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">z</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"> ( <font color="Maroon" title="c2">z</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">(<span class="default"><a href="binop_2.html#K9" title="BINOP_2:func.9">addreal</a> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a></span>)</span> implies <span class="p1">(<span class="default"><a href="binop_2.html#K9" title="BINOP_2:func.9">addreal</a> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K15" title="BINOP_2:func.15">addrat</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">z</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:54_1"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
<font color="Maroon" title="c2">z</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">(<span class="default"><a href="binop_2.html#K9" title="BINOP_2:func.9">addreal</a> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a></span>)</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"> <span class="p1">(<span class="default"><a href="binop_2.html#K9" title="BINOP_2:func.9">addreal</a> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K15" title="BINOP_2:func.15">addrat</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">z</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:54_1"/><span class="lab"><font color="Green" title="E23">A5</font></span>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  &amp; <font color="Maroon" title="c2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">x1</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c6">y1</font> = <font color="Maroon" title="c4">y</font> as   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_1"><span class="lab"><font color="Green" title="E23">A5</font></span></a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><a href="binop_2.html#K9" title="BINOP_2:func.9">addreal</a> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">z</font> = 
<a href="binop_2.html#K9" title="BINOP_2:func.9">addreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">y1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E3:54_1"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E3:54"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c5">x1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">y1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D9" onmouseover="rs('binop_2/D9')" onmouseout="rh()">BINOP_2:def 9</a></span>
<br/>.= 
<a href="binop_2.html#K15" title="BINOP_2:func.15">addrat</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">y1</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D15" onmouseover="rs('binop_2/D15')" onmouseout="rh()">BINOP_2:def 15</a></span>
<br/>.= 
<a href="binop_2.html#K15" title="BINOP_2:func.15">addrat</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_1"><span class="lab"><font color="Green" title="E23">A5</font></span></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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:54"/>
<a href="binop_2.html#K9" title="BINOP_2:func.9">addreal</a> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K15" title="BINOP_2:func.15">addrat</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E4:54"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="funct_1.html#T2" onmouseover="rs('funct_1/T2')" onmouseout="rh()">FUNCT_1:2</a>, <a class="ref" href="relat_1.html#T62" onmouseover="rs('relat_1/T62')" onmouseout="rh()">RELAT_1:62</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"> verum</span><br/>


</div>
