<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c2">x</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>



<b>thus </b><a NAME="E1:57"/>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><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><br/>

<b>assume </b><a NAME="E1:57_1"/><i><font color="Green" title="E42">A1</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c4">y</font> = <font color="Maroon" title="c3">y</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<b>consider </b><font color="Maroon" title="c5">z</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:57_1"/><i><font color="Green" title="E43">A2</font></i>: 
( <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c5">z</font> )
 <b>by </b><i><a class="txt" href="measure6.html#E1:57_1"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="measure6.html#D6" target="_self" title="MEASURE6:def.6">Def6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">t</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E6:57_1"/><i><font color="Green" title="E44">A3</font></i>: 
( <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c6">t</font> )
 <b>by </b><i><a class="txt" href="measure6.html#E4:57_1"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="measure6.html#D6" target="_self" title="MEASURE6:def.6">Def6</a></i>;<br/>
<b>thus </b><a NAME="E7:57_1"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="measure6.html#E4:57_1"><i><font color="Green" title="E43">A2</font></i></a>, <a class="txt" href="measure6.html#E6:57_1"><i><font color="Green" title="E44">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>

<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><br/>

<b>assume </b><a NAME="E2:57"/><i><font color="Green" title="E42">A4</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c4">y</font> = <font color="Maroon" title="c3">y</font> as  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">t</font> = <font color="Maroon" title="c4">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">x</font></span>)</span> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">z</font> = <font color="Maroon" title="c5">t</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E6:57"/>
<font color="Maroon" title="c5">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">y</font>
 
;<br/>
<a NAME="E7:57"/><b>then </b><i><font color="Green" title="E43">A5</font></i>: 
<font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="measure6.html#E2:57"><i><font color="Green" title="E42">A4</font></i></a>, <a class="ref" href="measure6.html#D6" target="_self" title="MEASURE6:def.6">Def6</a></i>;<br/>
<a NAME="E8:57"/>
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c5">t</font>
 
;<br/>
<b>hence </b><a NAME="E9:57"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="measure6.html#E7:57"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="measure6.html#D6" target="_self" title="MEASURE6:def.6">Def6</a></i>;<br/>


</div>
