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(^ Amalgamation ^)

[- Cross union -]

({ The thing whose features are all the unions of a feature of +a and a feature of +b will be called the cross union of +a and +b, and represented by +xu// +a +b. })

+Df +Xu [a [b [c = +ft// +c +xu// +a +b ]d ]f ( +ft// +d +a ( +ft// +f +b +sm// +c +un// +d +f
+Xu1 +Pr ) - +Xu = +ft// +c +xu// +a +b ]d ]f ( +ft// +d +a ( +ft// +f +b +sm// +c +un// +d +f

({ +xu// +a +b is symmetrical. })

+Xu2 +Pr ) - ( +Th ( +Am +Xu +sm// +xu// +a +b +xu// +b +a
+Xu3 +Pr ) - ( +Th ( +Am +Xu ) - +ft/ +b = +ft// +c +xu// +a +th/ +b ]d ( +ft// +d +a +sm// +c +un// +d +b
+Xu4 +Pr ) - ( +Th ( +Am +Xu ) - ( +ft/ +a +ft/ +b = +ft// +c +xu// +th/ +a +th/ +b +sm// +c +un// +a +b
+Xu5 +Pr ) - ( +Th ( +Am +Xu ) - ( +ft/ +a +ft/ +b +ft// +un// +a +b +xu// +th/ +a +th/ +b

({ The union of two features is a feature. })

+Un6 +Pr ) - ( +Th ( +Am +Xu ) - ( +ft/ +a +ft/ +b +ft/ +un// +a +b

({ The following Fact is called the associative property of unions. })

+Un7 +Pr ) - ( +Th ( +Am ( +Xu ( +ft/ +a ( +ft/ +b +ft/ +c +sm// +un// +a +un// +b +c +un// +un// +a +b +c

({ If +a and +b are features, then +th// +a +b is a feature. })

+Th21 +Pr ) - ( +Th ( +Am ( +Dv +Xu ) - ( +ft/ +a +ft/ +b +ft/ +th// +a +b

({ The thing whose features are +a, +b, and +c will be represented by +th/// +a +b +c. })

+Df +th/// +a +b +c +un// +th/ +a +th// +b +c
+Th22 +Pr ) - ( +Th ( +Am ( +Dv +Xu ) - ( +ft/ +a ( +ft/ +b +ft/ +c = +ft// +d +th/// +a +b +c ) +sm// +d +a ) +sm// +d +b +sm// +d +c
+Th23 +Pr ) - ( +Th ( +Am ( +Dv +Xu ) - ( +ft/ +a ( +ft/ +b +ft/ +c +ft/ +th/// +a +b +c
+Th24 +Pr ) - ( +Th ( +Am ( +Dv ( +Xu ( +ft/ +a ( +ft/ +b +ft/ +c +sm// +th/// +a +b +c +un// +th// +a +b +th/ +c

({ The thing whose features are +a, +b, +c, and +d will be represented by +th//// +a +b +c +d. })

+Df +th//// +a +b +c +d +un// +th// +a +b +th// +c +d
+Th25 +Pr ) - ( +Th ( +Am ( +Dv ( +Xu ( +ft/ +a ( +ft/ +b ( +ft/ +c +ft/ +d = +ft// +e +th//// +a +b +c +d ) +sm// +e +a ) +sm// +e +b ) +sm// +e +c +sm// +e +d
+Sm9 +Pr = +sm// +a +b [e = +ft// +e +a +ft// +e +b
+Th26 +Pr ) - ( +Th ( +Am ( +Dv ( +Xu ( +ft/ +a ( +ft/ +b +ft/ +d ) - +sm// +b +c +sm// +th//// +a +b +c +d +th/// +a +b +d
+Th27 +Pr ) - ( +Th ( +Am ( +Dv +Xu ) - ( +ft/ +a ( +ft/ +b ( +ft/ +c ( +ft/ +d +ft/ +e ) - +sm// +th// +th/ +a +th// +e +b +th// +th/ +c +th// +e +d ( +sm// +a +c +sm// +b +d
+Th28 +Pr ) - ( +Nf ( +Th ( +Am ( +Dv +Xu ) - +ft/ +b +ft/ +th// +0 +th/ +b

({ A link from +a to +b is a thing that has a first end at +a and a second end at +b. The following Definition gives an example. })

+Df +lk// +a +b +th// +th/ +a +th// +0 +th/ +b

({ The relation of being the same is transmitted for links. })

+Lk1 +Pr ) - ( +Nf ( +Th ( +Am ( +Dv ( +Xu ( +ft/ +a ( +ft/ +b +sm// +a +d +sm// +lk// +a +b +lk// +d +b
+Lk2 +Pr ) - ( +Nf ( +Th ( +Am ( +Dv ( +Xu ( +ft/ +a ( +ft/ +b +sm// +a +d +sm// +lk// +b +a +lk// +b +d

({ If +a and +b are features, then +lk// +a +b is a feature. })

+Lk3 +Pr ) - ( +Nf ( +Th ( +Am ( +Dv +Xu ) - ( +ft/ +a +ft/ +b +ft/ +lk// +a +b

({ If two links are the same, then the first end of one is the same as the first end of the other, and the second end of one is the same as the second end of the other. })

+Lk4 +Pr ) - ( +Nf ( +Th ( +Am ( +Dv +Xu ) - ( +ft/ +a ( +ft/ +b ( +ft/ +d +ft/ +f ) - +sm// +lk// +a +b +lk// +d +f ( +sm// +a +d +sm// +b +f

(_ PartialDifference _)

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