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<span class="kw">let </span><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">X1</font>, <font color="Olive" title="b2">X2</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b3">f1</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">X1</font>,<font color="Maroon" title="c2">Y</font><br/>  for <font color="Olive" title="b4">f2</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b2">X2</font>,<font color="Maroon" title="c2">Y</font>  st <font color="Olive" title="b3">f1</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Olive" title="b2">X2</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b4">f2</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Olive" title="b2">X2</font></span>)</span> holds <br/>( ( <font color="Olive" title="b1">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Olive" title="b2">X2</font> implies <font color="Olive" title="b3">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Olive" title="b4">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b4">f2</font> ) &amp; ( <font color="Olive" title="b3">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Olive" title="b4">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b4">f2</font> implies <font color="Olive" title="b1">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Olive" title="b2">X2</font> ) &amp; ( <font color="Olive" title="b2">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Olive" title="b1">X1</font> implies <font color="Olive" title="b3">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Olive" title="b4">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b3">f1</font> ) &amp; ( <font color="Olive" title="b3">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Olive" title="b4">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b3">f1</font> implies <font color="Olive" title="b2">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Olive" title="b1">X1</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">X1</font>, <font color="Maroon" title="c4">X2</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f1</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c2">Y</font><br/>  for <font color="Olive" title="b2">f2</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">X2</font>,<font color="Maroon" title="c2">Y</font>  st <font color="Olive" title="b1">f1</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X2</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">f2</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X2</font></span>)</span> holds <br/>( ( <font color="Maroon" title="c3">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">X2</font> implies <font color="Olive" title="b1">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Olive" title="b2">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b2">f2</font> ) &amp; ( <font color="Olive" title="b1">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Olive" title="b2">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b2">f2</font> implies <font color="Maroon" title="c3">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">X2</font> ) &amp; ( <font color="Maroon" title="c4">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font> implies <font color="Olive" title="b1">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Olive" title="b2">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">f1</font> ) &amp; ( <font color="Olive" title="b1">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Olive" title="b2">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">f1</font> implies <font color="Maroon" title="c4">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">f1</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c2">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f2</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">X2</font>,<font color="Maroon" title="c2">Y</font>  st <font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X2</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">f2</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X2</font></span>)</span> holds <br/>( ( <font color="Maroon" title="c3">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">X2</font> implies <font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Olive" title="b1">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">f2</font> ) &amp; ( <font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Olive" title="b1">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">f2</font> implies <font color="Maroon" title="c3">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">X2</font> ) &amp; ( <font color="Maroon" title="c4">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font> implies <font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Olive" title="b1">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">f1</font> ) &amp; ( <font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Olive" title="b1">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">f1</font> implies <font color="Maroon" title="c4">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">f2</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">X2</font>,<font color="Maroon" title="c2">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X2</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">f2</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X2</font></span>)</span> implies ( ( <font color="Maroon" title="c3">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">X2</font> implies <font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c6">f2</font> ) &amp; ( <font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c6">f2</font> implies <font color="Maroon" title="c3">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">X2</font> ) &amp; ( <font color="Maroon" title="c4">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font> implies <font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">f1</font> ) &amp; ( <font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">f1</font> implies <font color="Maroon" title="c4">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font> ) ) )</span><br/>









<span class="kw">reconsider </span><font color="Maroon" title="c7">Y1</font> = <font color="Maroon" title="c3">X1</font>, <font color="Maroon" title="c8">Y2</font> = <font color="Maroon" title="c4">X2</font>, <font color="Maroon" title="c9">Y3</font> = <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font> as    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T2" title="TSEP_1:th.2">TSEP_1:2</a>, <a class="ref" href="tsep_1.html#T22" title="TSEP_1:th.22">TSEP_1:22</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:142"/><span class="lab"><font color="Green" title="E95">A1</font></span>: 
<font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X2</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">f2</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c3">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">X2</font> implies <font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c6">f2</font> ) &amp; ( <font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c6">f2</font> implies <font color="Maroon" title="c3">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">X2</font> ) &amp; ( <font color="Maroon" title="c4">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font> implies <font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">f1</font> ) &amp; ( <font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">f1</font> implies <font color="Maroon" title="c4">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font> ) )</span><br/>

<div><span class="lab"><font color="Green" title="E96">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="142_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:142_1"/>
<font color="Maroon" title="c3">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">X2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c6">f2</font></span><br/><a NAME="E2:142_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A3</font></span>: 
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">X2</font>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">X2</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T23" title="TSEP_1:th.23">TSEP_1:23</a></span>;<br/><a NAME="E3:142_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c4">X2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:142"><span class="lab"><font color="Green" title="E95">A1</font></span></a>, <a class="ref" href="tmap_1.html#D12" target="_self" title="TMAP_1:def.12">Def13</a></span>;<br/><a NAME="E4:142_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#D5" target="_self" title="TMAP_1:def.5">Def5</a></span>;<br/><a NAME="E5:142_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">Y3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:142_1"><span class="lab"><font color="Green" title="E97">A3</font></span></a></span>;<br/><a NAME="E6:142_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c6">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#D5" target="_self" title="TMAP_1:def.5">Def5</a></span>;<br/><span class="kw">hence </span><a NAME="E7:142_1"/>
<font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c6">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#T67" target="_self" title="TMAP_1:th.67">Th74</a></span>; <a class="txt" 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>
<div><span class="lab"><font color="Green" title="E97">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="142_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:142_2"/>
<font color="Maroon" title="c4">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">f1</font></span><br/><a NAME="E2:142_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A5</font></span>: 
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X1</font>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">X1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T23" title="TSEP_1:th.23">TSEP_1:23</a></span>;<br/><a NAME="E3:142_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">f1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:142"><span class="lab"><font color="Green" title="E95">A1</font></span></a>, <a class="ref" href="tmap_1.html#D12" target="_self" title="TMAP_1:def.12">Def13</a></span>;<br/><a NAME="E4:142_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">Y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#D5" target="_self" title="TMAP_1:def.5">Def5</a></span>;<br/><a NAME="E5:142_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">Y3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:142_2"><span class="lab"><font color="Green" title="E98">A5</font></span></a></span>;<br/><a NAME="E6:142_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">f1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#D5" target="_self" title="TMAP_1:def.5">Def5</a></span>;<br/><span class="kw">hence </span><a NAME="E7:142_2"/>
<font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">f1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#T67" target="_self" title="TMAP_1:th.67">Th74</a></span>; <a class="txt" 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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="142_3"><span class="kw">now </span></a><div class="add"><a NAME="E1:142_3"/><span class="lab"><font color="Green" title="E98">A6</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">X2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><span class="kw">assume </span><a NAME="E2:142_3"/>
<font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c6">f2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">X2</font></span><br/><a NAME="E3:142_3"/><span class="kw">then </span>
<font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">X2</font>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">X2</font> #)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:142_3"><span class="lab"><font color="Green" title="E98">A6</font></span></a>, <a class="ref" href="tsep_1.html#T5" title="TSEP_1:th.5">TSEP_1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E4:142_3"/>
<font color="Maroon" title="c3">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">X2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T23" title="TSEP_1:th.23">TSEP_1:23</a></span>; <a class="txt" 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">hence </span><a NAME="E6:142"/>
( <font color="Maroon" title="c3">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">X2</font> iff <font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c6">f2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:142"><span class="lab"><font color="Green" title="E96">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font> iff <font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">f1</font> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="142_4"><span class="kw">now </span></a><div class="add"><a NAME="E1:142_4"/><span class="lab"><font color="Green" title="E98">A7</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><span class="kw">assume </span><a NAME="E2:142_4"/>
<font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">f1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font></span><br/><a NAME="E3:142_4"/><span class="kw">then </span>
<font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X1</font>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">X1</font> #)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:142_4"><span class="lab"><font color="Green" title="E98">A7</font></span></a>, <a class="ref" href="tsep_1.html#T5" title="TSEP_1:th.5">TSEP_1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E4:142_4"/>
<font color="Maroon" title="c4">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T23" title="TSEP_1:th.23">TSEP_1:23</a></span>; <a class="txt" 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">hence </span><a NAME="E8:142"/>
( <font color="Maroon" title="c4">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font> iff <font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">f1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:142"><span class="lab"><font color="Green" title="E97">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
