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<b>let </b><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#K4" title="NUMBERS:func.4">INT</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:4_1_1_1"/><i><font color="Green" title="E5">A1</font></i>: 
<font color="Maroon" title="c1">b</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a> 
 ;<br/><b>let </b><font color="Maroon" title="c2">i1</font>, <font color="Maroon" title="c3">i2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ;<br/><a NAME="E2:4_1_1_1"/>
( <font color="Maroon" title="c2">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  &amp; <font color="Maroon" title="c3">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  )
 ;<br/><a NAME="E3:4_1_1_1"/><b>then </b><i><font color="Green" title="E6">A2</font></i>: 
( <font color="Maroon" title="c2">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c3">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 <b>by </b><i><a class="ref" href="numbers.html#T15" title="NUMBERS:th.15">NUMBERS:15</a></i>;<br/><b>thus </b><font color="Maroon" title="c1">b</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c3">i2</font> = 
<font color="Maroon" title="c2">i1</font> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <font color="Maroon" title="c3">i2</font>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E1:4_1_1_1"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="binop_2.html#D20" title="BINOP_2:def.20">BINOP_2:def 20</a></i>
<br/>.= 
<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c3">i2</font>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E3:4_1_1_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="binop_2.html#D9" title="BINOP_2:def.9">BINOP_2:def 9</a></i>
;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:4_1_1"/><i><font color="Green" title="E5">A3</font></i>: 
for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">i2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  holds  <font color="Maroon" title="c1">b</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</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="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">i1</font>, <font color="Maroon" title="c3">i2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ;<br/><a NAME="E1:4_1_1_2"/>
( <font color="Maroon" title="c2">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  &amp; <font color="Maroon" title="c3">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  )
 ;<br/><a NAME="E2:4_1_1_2"/><b>then </b><i><font color="Green" title="E6">A4</font></i>: 
( <font color="Maroon" title="c2">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c3">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 <b>by </b><i><a class="ref" href="numbers.html#T15" title="NUMBERS:th.15">NUMBERS:15</a></i>;<br/><b>thus </b><font color="Maroon" title="c1">b</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c3">i2</font> = 
<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c3">i2</font>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E2:4_1_1"><i><font color="Green" title="E5">A3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">i1</font> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <font color="Maroon" title="c3">i2</font>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E2:4_1_1_2"><i><font color="Green" title="E6">A4</font></i></a>, <a class="ref" href="binop_2.html#D9" title="BINOP_2:def.9">BINOP_2:def 9</a></i>
<br/>.= 
<a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c3">i2</font>
<b>by </b><i><a class="ref" href="binop_2.html#D20" title="BINOP_2:def.20">BINOP_2:def 20</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:4_1_1"/>
<font color="Maroon" title="c1">b</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a> 
 <b>by </b><i><a class="ref" href="binop_1.html#T2" title="BINOP_1:th.2">BINOP_1:2</a></i>;<br/>


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