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<span class="kw">let </span><font color="Maroon" title="c1">b</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</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"> ( <font color="Maroon" title="c1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K34" title="BINOP_2:func.34">diffreal</a>  iff <font color="Maroon" title="c1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="finseqop.html#K8" title="FINSEQOP:func.8">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span>,<a href="binop_2.html#K31" title="BINOP_2:func.31">compreal</a>  )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">r1</font>, <font color="Maroon" title="c3">r2</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"> <a href="binop_2.html#K34" title="BINOP_2:func.34">diffreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">r1</font>,<font color="Maroon" title="c3">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="finseqop.html#K8" title="FINSEQOP:func.8">*</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span>,<a href="binop_2.html#K31" title="BINOP_2:func.31">compreal</a> </span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">r1</font>,<font color="Maroon" title="c3">r2</font></span><br/><span class="kw">thus </span><a href="binop_2.html#K34" title="BINOP_2:func.34">diffreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">r1</font>,<font color="Maroon" title="c3">r2</font> = 
<font color="Maroon" title="c2">r1</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c3">r2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D10" title="BINOP_2:def.10">BINOP_2:def 10</a></span>
<br/>.= 
<font color="Maroon" title="c2">r1</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">r2</font></span>)</span>

<br/>.= 
<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">r1</font>,<span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">r2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D9" title="BINOP_2:def.9">BINOP_2:def 9</a></span>
<br/>.= 
<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">r1</font>,<span class="p1">(<span class="default"><a href="binop_2.html#K31" title="BINOP_2:func.31">compreal</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">r2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D7" title="BINOP_2:def.7">BINOP_2:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="finseqop.html#K8" title="FINSEQOP:func.8">*</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span>,<a href="binop_2.html#K31" title="BINOP_2:func.31">compreal</a> </span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">r1</font>,<font color="Maroon" title="c3">r2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseqop.html#T87" title="FINSEQOP:th.87">FINSEQOP:87</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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:13_1_1"/>
( <font color="Maroon" title="c1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K34" title="BINOP_2:func.34">diffreal</a>  iff <font color="Maroon" title="c1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="finseqop.html#K8" title="FINSEQOP:func.8">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span>,<a href="binop_2.html#K31" title="BINOP_2:func.31">compreal</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#T2" title="BINOP_1:th.2">BINOP_1:2</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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