begin
theorem
canceled;
theorem
:: deftheorem MSAFREE3:def 1 :
canceled;
:: deftheorem Def2 defines Free MSAFREE3:def 2 :
for S being non void Signature
for X being ManySortedSet of the carrier of S
for b3 being strict MSAlgebra of S holds
( b3 = Free (S,X) iff ex A being MSSubset of (FreeMSA (X \/ ( the carrier of S --> {0}))) st
( b3 = GenMSAlg A & A = (Reverse (X \/ ( the carrier of S --> {0}))) "" X ) );
theorem Th3:
theorem Th4:
theorem Th5:
theorem Th6:
theorem
theorem Th8:
theorem Th9:
:: deftheorem Def3 defines variables_in MSAFREE3:def 3 :
for S being ManySortedSign
for t being non empty Relation
for b3 being ManySortedSet of the carrier of S holds
( b3 = S variables_in t iff for s being set st s in the carrier of S holds
b3 . s = { (a `1) where a is Element of rng t : a `2 = s } );
:: deftheorem defines variables_in MSAFREE3:def 4 :
for S being ManySortedSign
for X being ManySortedSet of the carrier of S
for t being non empty Relation holds X variables_in t = X /\ (S variables_in t);
theorem Th10:
theorem Th11:
theorem Th12:
theorem Th13:
theorem Th14:
theorem Th15:
:: deftheorem defines variables_in MSAFREE3:def 5 :
for S being non void Signature
for X being V5() ManySortedSet of the carrier of S
for t being Term of S,X holds variables_in t = S variables_in t;
theorem Th16:
:: deftheorem Def6 defines -Terms MSAFREE3:def 6 :
for S being non void Signature
for Y being V5() ManySortedSet of the carrier of S
for X being ManySortedSet of the carrier of S
for b4 being MSSubset of (FreeMSA Y) holds
( b4 = S -Terms (X,Y) iff for s being SortSymbol of S holds b4 . s = { t where t is Term of S,Y : ( the_sort_of t = s & variables_in t c= X ) } );
theorem Th17:
theorem Th18:
theorem Th19:
theorem Th20:
theorem Th21:
theorem Th22:
theorem Th23:
theorem Th24:
theorem Th25:
theorem
theorem Th27:
theorem Th28:
theorem Th29:
theorem Th30:
theorem Th31:
theorem
theorem Th33:
theorem Th34:
theorem Th35:
theorem
theorem
theorem Th38:
theorem Th39:
theorem Th40: