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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being  <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a><br/> for <font color="Olive" title="b2">seq1</font>, <font color="Olive" title="b3">seq2</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Olive" title="b1">X</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Olive" title="b2">seq1</font></span>)</span></span>)</span> <a href="bhsp_1.html#K8" title="BHSP_1:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Olive" title="b3">seq2</font></span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <font color="Olive" title="b2">seq1</font></span>)</span> <a href="bhsp_1.html#K8" title="BHSP_1:func.8">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <font color="Olive" title="b3">seq2</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">X</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">seq1</font>, <font color="Olive" title="b2">seq2</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c3">X</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Olive" title="b1">seq1</font></span>)</span></span>)</span> <a href="bhsp_1.html#K8" title="BHSP_1:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Olive" title="b2">seq2</font></span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <font color="Olive" title="b1">seq1</font></span>)</span> <a href="bhsp_1.html#K8" title="BHSP_1:func.8">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <font color="Olive" title="b2">seq2</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">seq1</font>, <font color="Maroon" title="c5">seq2</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c3">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span> <a href="bhsp_1.html#K8" title="BHSP_1:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c5">seq2</font></span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <font color="Maroon" title="c4">seq1</font></span>)</span> <a href="bhsp_1.html#K8" title="BHSP_1:func.8">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <font color="Maroon" title="c5">seq2</font></span>)</span></span>)</span></span><br/>









<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span> <a href="bhsp_1.html#K8" title="BHSP_1:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c5">seq2</font></span>)</span></span>)</span> = 
<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"><font color="Maroon" title="c1">a</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span> <a href="bhsp_1.html#K8" title="BHSP_1:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c5">seq2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="bhsp_4.html#T3" target="_self" title="BHSP_4:th.3">Th3</a></i>
<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"><font color="Maroon" title="c1">a</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span> <a href="bhsp_1.html#K8" title="BHSP_1:func.8">+</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"><font color="Maroon" title="c2">b</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <font color="Maroon" title="c5">seq2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="bhsp_4.html#T3" target="_self" title="BHSP_4:th.3">Th3</a></i>
<br/>.= 
 <a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <font color="Maroon" title="c4">seq1</font></span>)</span> <a href="bhsp_1.html#K8" title="BHSP_1:func.8">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <font color="Maroon" title="c5">seq2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="bhsp_4.html#T1" target="_self" title="BHSP_4:th.1">Th1</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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