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<span class="kw">let </span><font color="Maroon" title="c1">I</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Olive" title="b1">G</font><br/>  for <font color="Olive" title="b3">x</font> being  <font color="Maroon" title="c1">I</font> <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><br/>  for <font color="Olive" title="b4">x0</font> being  <font color="Maroon" title="c1">I</font> <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive">b<sub>2</sub></font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b3">x</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b4">x0</font> holds <br/> <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Olive" title="b4">x0</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">H</font> being    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font><br/>  for <font color="Olive" title="b2">x</font> being  <font color="Maroon" title="c1">I</font> <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><br/>  for <font color="Olive" title="b3">x0</font> being  <font color="Maroon" title="c1">I</font> <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b2">x</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b3">x0</font> holds <br/> <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Olive" title="b3">x0</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">H</font> be    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being  <font color="Maroon" title="c1">I</font> <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><br/>  for <font color="Olive" title="b2">x0</font> being  <font color="Maroon" title="c1">I</font> <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">x</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b2">x0</font> holds <br/> <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Olive" title="b2">x0</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">x</font> be  <font color="Maroon" title="c1">I</font> <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x0</font> being  <font color="Maroon" title="c1">I</font> <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Maroon" title="c4">x</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b1">x0</font> holds <br/> <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Olive" title="b1">x0</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">x0</font> be  <font color="Maroon" title="c1">I</font> <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c5">x0</font> implies  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Maroon" title="c5">x0</font> )</span><br/>









<span class="kw">assume </span><a NAME="E1:40"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c5">x0</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Maroon" title="c5">x0</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">f</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">G</font><span class="kw"> such that </span><br/><a NAME="E3:40"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
(  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <font color="Maroon" title="c1">I</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_17.html#D1" target="_self" onmouseover="rs('group_17/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">g</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H</font><span class="kw"> such that </span><br/><a NAME="E5:40"/><span class="lab"><font color="Green" title="E43">A3</font></span>: 
(  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Maroon" title="c5">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c7">g</font> &amp; <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x0</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <font color="Maroon" title="c1">I</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_17.html#D1" target="_self" onmouseover="rs('group_17/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<span class="kw">thus </span><a NAME="E6:40"/>
 <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Maroon" title="c5">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="group_17.html#T31" target="_self" onmouseover="rs('group_17/T31')" onmouseout="rh()">Th31</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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