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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="lopban_2.html#NM3" title="LOPBAN_2:NM.3">Banach_Algebra</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">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">z</font>, <font color="Olive" title="b3">w</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">z</font>,<font color="Olive" title="b3">w</font> <a href="lopban_4.html#R1" title="LOPBAN_4:pred.1">are_commutative</a>  holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Olive" title="b2">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Olive" title="b3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b2">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b3">w</font></span>)</span> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> <font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">z</font>,<font color="Olive" title="b3">w</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">k</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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">z</font>, <font color="Olive" title="b2">w</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">w</font> <a href="lopban_4.html#R1" title="LOPBAN_4:pred.1">are_commutative</a>  holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Olive" title="b2">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">w</font></span>)</span> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> <font color="Maroon" title="c2">k</font>,<font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">w</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">z</font>, <font color="Maroon" title="c4">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</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="c3">z</font>,<font color="Maroon" title="c4">w</font> <a href="lopban_4.html#R1" title="LOPBAN_4:pred.1">are_commutative</a>  implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:39"/>
<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font> <a href="lopban_4.html#R1" title="LOPBAN_4:pred.1">are_commutative</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span><br/>

<span class="kw">then </span><span class="lab"><font color="Green" title="E38">A1</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="lopban_4.html#K7" title="LOPBAN_4:func.7">Alfa</a> <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_4.html#T25" target="_self" title="LOPBAN_4:th.25">Th25</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="lopban_4.html#K7" title="LOPBAN_4:func.7">Alfa</a> <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_3.html#D10" title="LOPBAN_3:def.10">LOPBAN_3:def 10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="lopban_4.html#K7" title="LOPBAN_4:func.7">Alfa</a> <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_4.html#T9" target="_self" title="LOPBAN_4:th.9">Th9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="lopban_4.html#K7" title="LOPBAN_4:func.7">Alfa</a> <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#D6" title="NORMSP_1:def.6">NORMSP_1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p3">(<span class="default"><a href="lopban_4.html#K7" title="LOPBAN_4:func.7">Alfa</a> <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_3.html#T21" title="LOPBAN_3:th.21">LOPBAN_3:21</a></span>
;<br/>
<a NAME="E3:39"/>
for <font color="Olive" title="b1">l</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">k</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K7" title="LOPBAN_4:func.7">Alfa</a> <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">l</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">l</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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="c5">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">k</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K7" title="LOPBAN_4:func.7">Alfa</a> <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">l</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:39_2"/><span class="lab"><font color="Green" title="E39">A2</font></span>: 
<font color="Maroon" title="c5">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K7" title="LOPBAN_4:func.7">Alfa</a> <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">l</font></span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K7" title="LOPBAN_4:func.7">Alfa</a> <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">l</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K7" title="LOPBAN_4:func.7">Alfa</a> <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">l</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#D6" title="NORMSP_1:def.6">NORMSP_1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K7" title="LOPBAN_4:func.7">Alfa</a> <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">l</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_3.html#D10" title="LOPBAN_3:def.10">LOPBAN_3:def 10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">l</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_2"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="ref" href="lopban_4.html#D8" target="_self" title="LOPBAN_4:def.8">Def8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">l</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"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">l</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">l</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_2"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="ref" href="lopban_4.html#D9" target="_self" title="LOPBAN_4:def.9">Def9</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="E4:39"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="lopban_4.html#T11" target="_self" title="LOPBAN_4:th.11">Th11</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>
