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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zmodul02.html#K11" title="ZMODUL02:func.11">LinComb</a> <font color="Maroon" title="c2">V</font>,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <span class="p1">(<span class="default"><a href="zmodul02.html#K12" title="ZMODUL02:func.12">@</a> $2</span>)</span> );<br/>
<a NAME="E1:34_1_1"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><br/>  for <font color="Olive" title="b2">e1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zmodul02.html#K11" title="ZMODUL02:func.11">LinComb</a> <font color="Maroon" title="c2">V</font>  ex <font color="Olive" title="b3">e2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zmodul02.html#K11" title="ZMODUL02:func.11">LinComb</a> <font color="Maroon" title="c2">V</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">e1</font>,<font color="Olive" title="b3">e2</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="34_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</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">e1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zmodul02.html#K11" title="ZMODUL02:func.11">LinComb</a> <font color="Maroon" title="c2">V</font>  ex <font color="Olive" title="b2">e2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zmodul02.html#K11" title="ZMODUL02:func.11">LinComb</a> <font color="Maroon" title="c2">V</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">e1</font>,<font color="Olive" title="b2">e2</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c4">e1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zmodul02.html#K11" title="ZMODUL02:func.11">LinComb</a> <font color="Maroon" title="c2">V</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">  ex <font color="Olive" title="b1">e2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zmodul02.html#K11" title="ZMODUL02:func.11">LinComb</a> <font color="Maroon" title="c2">V</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">e1</font>,<font color="Olive" title="b1">e2</font>]</span><br/>



<span class="kw">take </span>
 <a href="zmodul02.html#K13" title="ZMODUL02:func.13">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <span class="p2">(<span class="default"><a href="zmodul02.html#K12" title="ZMODUL02:func.12">@</a> <font color="Maroon" title="c4">e1</font></span>)</span></span>)</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"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">e1</font>, <a href="zmodul02.html#K13" title="ZMODUL02:func.13">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <span class="p2">(<span class="default"><a href="zmodul02.html#K12" title="ZMODUL02:func.12">@</a> <font color="Maroon" title="c4">e1</font></span>)</span></span>)</span>]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c3">x</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"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp;  <a href="zmodul02.html#K13" title="ZMODUL02:func.13">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <span class="p2">(<span class="default"><a href="zmodul02.html#K12" title="ZMODUL02:func.12">@</a> <font color="Maroon" title="c4">e1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <span class="p1">(<span class="default"><a href="zmodul02.html#K12" title="ZMODUL02:func.12">@</a> <font color="Maroon" title="c4">e1</font></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E1:34_1_1_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp;  <a href="zmodul02.html#K13" title="ZMODUL02:func.13">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <span class="p2">(<span class="default"><a href="zmodul02.html#K12" title="ZMODUL02:func.12">@</a> <font color="Maroon" title="c4">e1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <span class="p1">(<span class="default"><a href="zmodul02.html#K12" title="ZMODUL02:func.12">@</a> <font color="Maroon" title="c4">e1</font></span>)</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/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c3">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<span class="p2">(<span class="default"><a href="zmodul02.html#K11" title="ZMODUL02:func.11">LinComb</a> <font color="Maroon" title="c2">V</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p1">(<span class="default"><a href="zmodul02.html#K11" title="ZMODUL02:func.11">LinComb</a> <font color="Maroon" title="c2">V</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:34_1_1"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><br/>  for <font color="Olive" title="b2">e</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zmodul02.html#K11" title="ZMODUL02:func.11">LinComb</a> <font color="Maroon" title="c2">V</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">e</font>,<font color="Maroon" title="c3">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">e</font>)]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="binop_1.html#S3" onmouseover="rs('binop_1/S3')" onmouseout="rh()">BINOP_1:sch 3</a>(<a class="txt" href="#E1:34_1_1"><span class="lab"><font color="Green" title="E17">A1</font></span></a>);<br/></span>
<span class="kw">take </span>
<font color="Maroon" title="c3">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">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><br/>  for <font color="Olive" title="b2">e</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zmodul02.html#K11" title="ZMODUL02:func.11">LinComb</a> <font color="Maroon" title="c2">V</font> holds  <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">e</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <span class="p1">(<span class="default"><a href="zmodul02.html#K12" title="ZMODUL02:func.12">@</a> <font color="Olive" title="b2">e</font></span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</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">e</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zmodul02.html#K11" title="ZMODUL02:func.11">LinComb</a> <font color="Maroon" title="c2">V</font> holds  <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Olive" title="b1">e</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <span class="p1">(<span class="default"><a href="zmodul02.html#K12" title="ZMODUL02:func.12">@</a> <font color="Olive" title="b1">e</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">e</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zmodul02.html#K11" title="ZMODUL02:func.11">LinComb</a> <font color="Maroon" title="c2">V</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"> <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">e</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <span class="p1">(<span class="default"><a href="zmodul02.html#K12" title="ZMODUL02:func.12">@</a> <font color="Maroon" title="c5">e</font></span>)</span></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c6">a0</font> = <font color="Maroon" title="c4">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> ;<br/>
<a NAME="E5:34_1_1"/>
 ex <font color="Olive" title="b1">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a0</font> &amp; <font color="Maroon" title="c3">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c6">a0</font>,<font color="Maroon" title="c5">e</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <span class="p1">(<span class="default"><a href="zmodul02.html#K12" title="ZMODUL02:func.12">@</a> <font color="Maroon" title="c5">e</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_1_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:34_1_1"/>
<font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">e</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <span class="p1">(<span class="default"><a href="zmodul02.html#K12" title="ZMODUL02:func.12">@</a> <font color="Maroon" title="c5">e</font></span>)</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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