<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">j</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</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">G</font> being  <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a><br/> for <font color="Olive" title="b2">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">G</font> holds  <font color="Olive" title="b2">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c2">j</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</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">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">G</font> holds  <font color="Olive" title="b1">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c2">j</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">G</font>; <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="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c2">j</font></span>)</span></span><br/>







<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>] <span class="kw">means </span>for <font color="Olive" title="b1">i</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> holds  <font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> $1</span>)</span>;<br/>
<a NAME="E1:83"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
for <font color="Olive" title="b1">j</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>] holds <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> 1] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">j</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</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">S<sub>1</sub></font>[<font color="Maroon" title="c5">j</font>] implies ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">j</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">j</font> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> 1] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:83_1"/><span class="lab"><font color="Green" title="E47">A2</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> holds  <font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c5">j</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">j</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">j</font> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> 1] )</span><br/>

<span class="kw">thus </span><a NAME="E2:83_1"/>
for <font color="Olive" title="b1">i</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> holds  <font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">j</font> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> 1]</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</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="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> = 
<font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <font color="Maroon" title="c5">j</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c5">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_1"><span class="lab"><font color="Green" title="E47">A2</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c5">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="group_1.html#E86"><span class="lab"><font color="Green" title="E42">Lm14</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c5">j</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" target="_self" title="GROUP_1:def.4">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_1"><span class="lab"><font color="Green" title="E47">A2</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</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">let </span><font color="Maroon" title="c6">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</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="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> 1</span>)</span></span>)</span></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> 1</span>)</span></span>)</span> = 
<font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> 1</span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <font color="Maroon" title="c5">j</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> 1</span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c5">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_1"><span class="lab"><font color="Green" title="E47">A2</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> 1</span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c5">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="group_1.html#E88"><span class="lab"><font color="Green" title="E44">Lm16</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> 1</span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c5">j</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" target="_self" title="GROUP_1:def.4">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_1"><span class="lab"><font color="Green" title="E47">A2</font></span></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>
<a NAME="E2:83"/><span class="lab"><font color="Green" title="E47">A3</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</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="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">G</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D5" target="_self" title="GROUP_1:def.5">Def5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D8" target="_self" title="GROUP_1:def.8">Def8</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>
<a NAME="E3:83"/>
for <font color="Olive" title="b1">j</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="int_1.html#S4" title="INT_1:sch.4">INT_1:sch 4</a>(<a class="txt" href="#E2:83"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E46">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:83"/>
<font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c2">j</font></span>)</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>
