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<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">z</font>, <font color="Maroon" title="c4">w</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</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">vx</font>, <font color="Olive" title="b2">vy</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2  st <font color="Olive" title="b1">vx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Olive" title="b2">vy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> holds <br/>( <font color="Olive" title="b1">vx</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Olive" title="b2">vy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">z</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">w</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Olive" title="b1">vx</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Olive" title="b2">vy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">z</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">w</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">vx</font>, <font color="Maroon" title="c6">vy</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2; <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="c5">vx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c6">vy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> implies ( <font color="Maroon" title="c5">vx</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c6">vy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">z</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">w</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c5">vx</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">vy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">z</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">w</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> ) )</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c7">x1</font> = <font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c8">y1</font> = <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c9">z1</font> = <font color="Maroon" title="c3">z</font>, <font color="Maroon" title="c10">w1</font> = <font color="Maroon" title="c4">w</font> as    <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">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:9"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
( <font color="Maroon" title="c5">vx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c6">vy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&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="c5">vx</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c6">vy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">z</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">w</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c5">vx</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">vy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">z</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">w</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> )</span><br/>

<span class="kw">hence </span><font color="Maroon" title="c5">vx</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c6">vy</font> = 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K9" title="BINOP_2:func.9">addreal</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c9">z1</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="binop_2.html#K9" title="BINOP_2:func.9">addreal</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c8">y1</font>,<font color="Maroon" title="c10">w1</font>)</span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T75" onmouseover="rs('finseq_2/T75')" onmouseout="rh()">FINSEQ_2:75</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c9">z1</font></span>)</span>,<span class="p2">(<span class="default"><a href="binop_2.html#K9" title="BINOP_2:func.9">addreal</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c8">y1</font>,<font color="Maroon" title="c10">w1</font>)</span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<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/>.= 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">z</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">w</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<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 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="c5">vx</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">vy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">z</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">w</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c5">vx</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">vy</font> = 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K10" title="BINOP_2:func.10">diffreal</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c9">z1</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="binop_2.html#K10" title="BINOP_2:func.10">diffreal</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c8">y1</font>,<font color="Maroon" title="c10">w1</font>)</span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="finseq_2.html#T75" onmouseover="rs('finseq_2/T75')" onmouseout="rh()">FINSEQ_2:75</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c9">z1</font></span>)</span>,<span class="p2">(<span class="default"><a href="binop_2.html#K10" title="BINOP_2:func.10">diffreal</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c8">y1</font>,<font color="Maroon" title="c10">w1</font>)</span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D10" onmouseover="rs('binop_2/D10')" onmouseout="rh()">BINOP_2:def 10</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">z</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">w</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D10" onmouseover="rs('binop_2/D10')" onmouseout="rh()">BINOP_2:def 10</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/>


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