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<span class="kw">let </span><font color="Maroon" title="c1">a9</font>, <font color="Maroon" title="c2">b9</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</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">a</font>, <font color="Olive" title="b2">b</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c1">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Maroon" title="c2">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> holds <br/> <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c1">a9</font>,<font color="Maroon" title="c2">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">b</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</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> ; <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="c1">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c2">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> implies  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c1">a9</font>,<font color="Maroon" title="c2">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">b</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:3"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
<font color="Maroon" title="c1">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">and </span><br/><a NAME="E2:3"/><span class="lab"><font color="Green" title="E5">A2</font></span>: 
<font color="Maroon" title="c2">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c1">a9</font>,<font color="Maroon" title="c2">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">b</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c6">x2</font>, <font color="Maroon" title="c7">y1</font>, <font color="Maroon" title="c8">y2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E4:3"/><span class="lab"><font color="Green" title="E6">A3</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <span class="kw">and </span><br/><a NAME="E5:3"/><span class="lab"><font color="Green" title="E7">A4</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c7">y1</font>,<font color="Maroon" title="c8">y2</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <span class="kw">and </span><br/><a NAME="E6:3"/><span class="lab"><font color="Green" title="E8">A5</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c7">y1</font></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c8">y2</font></span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D4" title="XCMPLX_0:def.4">XCMPLX_0:def 4</a></span>;<br/>
<a NAME="E7:3"/><span class="lab"><font color="Green" title="E9">A6</font></span>: 
<font color="Maroon" title="c8">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:3"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="txt" href="xreal_1.html#E2"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a></span>;<br/>
<a NAME="E8:3"/>
<font color="Maroon" title="c6">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:3"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="xreal_1.html#E2"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a></span>;<br/>
<a NAME="E9:3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A7</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c8">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:3"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="ref" href="arytm_0.html#T13" title="ARYTM_0:th.13">ARYTM_0:13</a></span>;<br/>
<a NAME="E10:3"/><span class="lab"><font color="Green" title="E11">A8</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:3"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="txt" href="xreal_1.html#E2"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a></span>;<br/>
<a NAME="E11:3"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:3"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="xreal_1.html#E2"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:3"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c1">a9</font>,<font color="Maroon" title="c2">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E2:3"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E6:3"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="txt" href="#E10:3"><span class="lab"><font color="Green" title="E11">A8</font></span></a>, <a class="txt" href="#E9:3"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="ref" href="arytm_0.html#D7" title="ARYTM_0:def.7">ARYTM_0:def 7</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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