begin
theorem Th1:
theorem Th2:
theorem
theorem Th4:
:: deftheorem Def1 defines <= MSUHOM_1:def 1 :
for S, S9 being non empty ManySortedSign holds
( S <= S9 iff ( the carrier of S c= the carrier of S9 & the carrier' of S c= the carrier' of S9 & the Arity of S9 | the carrier' of S = the Arity of S & the ResultSort of S9 | the carrier' of S = the ResultSort of S ) );
theorem
theorem
theorem
theorem Th8:
:: deftheorem Def2 defines Over MSUHOM_1:def 2 :
for S, S9 being non empty non void strict ManySortedSign
for A being strict non-empty MSAlgebra of S9 st S <= S9 holds
for b4 being strict non-empty MSAlgebra of S holds
( b4 = A Over S iff ( the Sorts of b4 = the Sorts of A | the carrier of S & the Charact of b4 = the Charact of A | the carrier' of S ) );
theorem Th9:
theorem Th10:
:: deftheorem Def3 defines MSAlg MSUHOM_1:def 3 :
for U1, U2 being Universal_Algebra
for h being Function of U1,U2 st MSSign U1 = MSSign U2 holds
MSAlg h = 0 .--> h;
theorem Th11:
theorem Th12:
Lm1:
for U1 being Universal_Algebra holds dom (signature U1) = dom the charact of U1
theorem Th13:
Lm2:
for U1, U2 being Universal_Algebra st U1,U2 are_similar holds
for o being OperSymbol of (MSSign U1) holds Den (o,((MSAlg U2) Over (MSSign U1))) is operation of U2
theorem Th14:
theorem Th15:
theorem Th16:
Lm3:
for U1 being Universal_Algebra
for n being Nat st n in dom the charact of U1 holds
n is OperSymbol of (MSSign U1)
theorem Th17:
theorem Th18:
theorem Th19:
theorem
theorem Th21:
theorem Th22:
theorem Th23:
theorem
theorem
theorem