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<span class="kw">let </span><font color="Maroon" title="c1">C1</font>, <font color="Maroon" title="c2">C2</font> be   <a href="altcat_1.html#V6" title="ALTCAT_1:attr.6">strict</a>   <a href="altcat_1.html#L2" title="ALTCAT_1:struct.2">AltCatStr</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"> (  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; (  for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds <br/> (  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c1">C1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) &amp; (  for <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">j</font>, <font color="Olive" title="b5">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds   the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c1">C1</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> (<font color="Olive" title="b3">i</font>,<font color="Olive" title="b4">j</font>,<font color="Olive" title="b5">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> (<font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b3">i</font>,<font color="Olive" title="b4">j</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b4">j</font>,<font color="Olive" title="b5">k</font>)) ) ) ) &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; (  for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds <br/> (  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c2">C2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) &amp; (  for <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">j</font>, <font color="Olive" title="b5">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds   the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c2">C2</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> (<font color="Olive" title="b3">i</font>,<font color="Olive" title="b4">j</font>,<font color="Olive" title="b5">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> (<font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b3">i</font>,<font color="Olive" title="b4">j</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b4">j</font>,<font color="Olive" title="b5">k</font>)) ) ) ) implies <font color="Maroon" title="c1">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">C2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:34_1"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 <span class="kw">and </span><br/><a NAME="E2:34_1"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
 for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds <br/> (  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c1">C1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) &amp; (  for <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">j</font>, <font color="Olive" title="b5">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds   the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c1">C1</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> (<font color="Olive" title="b3">i</font>,<font color="Olive" title="b4">j</font>,<font color="Olive" title="b5">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> (<font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b3">i</font>,<font color="Olive" title="b4">j</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b4">j</font>,<font color="Olive" title="b5">k</font>)) ) )
 <span class="kw">and </span><br/><a NAME="E3:34_1"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 <span class="kw">and </span><br/><a NAME="E4:34_1"/><span class="lab"><font color="Green" title="E16">A4</font></span>: 
 for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds <br/> (  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c2">C2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) &amp; (  for <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">j</font>, <font color="Olive" title="b5">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds   the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c2">C2</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> (<font color="Olive" title="b3">i</font>,<font color="Olive" title="b4">j</font>,<font color="Olive" title="b5">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> (<font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b3">i</font>,<font color="Olive" title="b4">j</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b4">j</font>,<font color="Olive" title="b5">k</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="c1">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">C2</font></span><br/>

<div><span class="lab"><font color="Green" title="E17">A5</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="34_1_1"><span class="kw">now </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">  for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font>, <font color="Olive" title="b3">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c1">C1</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>,<font color="Olive" title="b3">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c2">C2</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>,<font color="Olive" title="b3">k</font>)</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c4">j</font>, <font color="Maroon" title="c5">k</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() implies  the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c1">C1</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c5">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c2">C2</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c5">k</font>) )</span><br/><span class="kw">assume </span><a NAME="E1:34_1_1"/><span class="lab"><font color="Green" title="E18">A6</font></span>: 
( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></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">  the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c1">C1</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c5">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c2">C2</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c5">k</font>)</span><br/><span class="kw">hence </span> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c1">C1</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c5">k</font>) = 
 <a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> (<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c5">k</font>))
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_1"><span class="lab"><font color="Green" title="E14">A2</font></span></a></span>
<br/>.= 
 the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c2">C2</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c5">k</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_1"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E1:34_1_1"><span class="lab"><font color="Green" title="E18">A6</font></span></a></span>
;<br/> <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>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="34_1_2"><span class="kw">now </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">  for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds   the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c1">C1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c2">C2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c4">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></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">  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c1">C1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c2">C2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span><br/><span class="kw">thus </span> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c1">C1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>) = 
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_1"><span class="lab"><font color="Green" title="E14">A2</font></span></a></span>
<br/>.= 
 the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c2">C2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_1"><span class="lab"><font color="Green" title="E16">A4</font></span></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/></div><span class="kw">end;</span></div>
<a NAME="E7:34_1"/><span class="kw">then </span>
 the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c1">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c2">C2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_1"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E3:34_1"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="ref" href="altcat_1.html#T7" onmouseover="rs('altcat_1/T7')" onmouseout="rh()">ALTCAT_1:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:34_1"/>
<font color="Maroon" title="c1">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_1"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E3:34_1"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E5:34_1"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="altcat_1.html#T8" onmouseover="rs('altcat_1/T8')" onmouseout="rh()">ALTCAT_1:8</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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