Equality

Two mathematical objects are equal if and only if they are precisely the same in every way. This defines a binary relation, equality, denoted by the sign of equality has symbol:: = in such a way that the statement "x = y" means that x and y are equal.

Some properties:

expression:= "all a . a = a" label:="reflexivity"

expression1:="all a,b . a = b -> b = a" label1:="symmetry"

expression2:="all a,b,c . a = b /\ b = c -> a = c"label2:="transitivity"

Where "/\" is logical (ML -style ) "and", "->" is implication and "all" is the Universal Quantifier

S-expressions for Equalitylabel:=* for example is S-expression:="(all (a) (= a a))"; for Equalitylabel1:=* S-expression:="(all (a b) (-> (= a b)(= b a)))"; etc

The substitution property states:

For any quantities a and b and any expression F(x), if a = b, then F(a) = F(b)

Some specific examples of this are:

expression3:="all a,b,c . a = b -> a / c = b / c)" (here F(x) is x / c); label3:="substitution /"

"all a,b,c . a = b -> a * c = b * c)" "all a,b,c . a = b -> a + c = b + c)" "all a,b,c . a = b -> a - c = b - c)"