
theorem Th28:
  for S1,S2 being non void non empty ManySortedSign for A1 being
non-empty MSAlgebra over S1, A2 being non-empty MSAlgebra over S2 st the Sorts
of A1 tolerates the Sorts of A2 & the Charact of A1 tolerates the Charact of A2
for o being OperSymbol of S1+*S2, o1 being OperSymbol of S1 st o = o1 holds Den
  (o, A1+*A2) = Den(o1, A1)
proof
  let S1,S2 be non void non empty ManySortedSign;
  let A1 be non-empty MSAlgebra over S1, A2 be non-empty MSAlgebra over S2;
A1: dom the Charact of A1 = the carrier' of S1 by PARTFUN1:def 2;
  assume the Sorts of A1 tolerates the Sorts of A2;
  then
A2: the Charact of A1+*A2 = (the Charact of A1)+*the Charact of A2 by Def4;
  assume
A3: the Charact of A1 tolerates the Charact of A2;
  let o be OperSymbol of S1+*S2, o1 be OperSymbol of S1;
  assume o = o1;
  hence thesis by A2,A3,A1,FUNCT_4:15;
end;
