:: deftheorem Def4 defines OSREL OSAFREE:def 4 :
for S being OrderSortedSign
for X being ManySortedSet of S
for b₃ being Relation of ([: the carrier' of S,{ the carrier of S}:] \/ (Union (coprod X))),(([: the carrier' of S,{ the carrier of S}:] \/ (Union (coprod X))) *) holds
( b₃ = OSREL X iff for a being Element of [: the carrier' of S,{ the carrier of S}:] \/ (Union (coprod X))
for b being Element of ([: the carrier' of S,{ the carrier of S}:] \/ (Union (coprod X))) * holds
( [a,b] in b₃ iff ( a in [: the carrier' of S,{ the carrier of S}:] & ( for o being OperSymbol of S st [o, the carrier of S] = a holds
( len b = len (the_arity_of o) & ( for x being set st x in dom b holds
( ( b . x in [: the carrier' of S,{ the carrier of S}:] implies for o1 being OperSymbol of S st [o1, the carrier of S] = b . x holds
the_result_sort_of o1 <= (the_arity_of o) /. x ) & ( b . x in Union (coprod X) implies ex i being Element of S st
( i <= (the_arity_of o) /. x & b . x in coprod (i,X) ) ) ) ) ) ) ) ) );