:: deftheorem Def7 defines REL MSAFREE:def 7 :
for S being non empty non void ManySortedSign
for X being ManySortedSet of the carrier 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 = REL 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 b . x in coprod (((the_arity_of o) . x),X) ) ) ) ) ) ) ) );