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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#K4" title="NUMBERS:func.4">INT</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K44" title="BINOP_2:func.44">addint</a>  iff  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#K4" title="BINOP_1:func.4">.</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>) )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  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#K4" title="BINOP_1:func.4">.</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>) ) implies <font color="Maroon" title="c1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:2_1_1_1"/><span class="lab"><font color="Green" title="E2">A1</font></span>: 
<font color="Maroon" title="c1">b</font> <a href="binop_1.html#R8" title="BINOP_1:pred.8">=</a>  <a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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#K4" title="BINOP_1:func.4">.</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>)</span><br/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">b</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c3">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="Maroon" title="c2">i1</font>,<font color="Maroon" title="c3">i2</font>)</span><br/><span class="kw">thus </span><font color="Maroon" title="c1">b</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_1"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="ref" href="binop_2.html#D20" title="BINOP_2:def.20">BINOP_2:def 20</a></span>
<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>)
<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>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:2_1_1"/><span class="lab"><font color="Green" title="E2">A2</font></span>: 
 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#K4" title="BINOP_1:func.4">.</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>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K44" title="BINOP_2:func.44">addint</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">b</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c3">i2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c3">i2</font>)</span><br/><span class="kw">thus </span><font color="Maroon" title="c1">b</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1"><span class="lab"><font color="Green" title="E2">A2</font></span></a></span>
<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>
<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#K44" title="BINOP_2:func.44">addint</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c3">i2</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D20" title="BINOP_2:def.20">BINOP_2:def 20</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:2_1_1"/>
<font color="Maroon" title="c1">b</font> <a href="binop_1.html#R8" title="BINOP_1:pred.8">=</a>  <a href="binop_2.html#K44" title="BINOP_2:func.44">addint</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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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