
theorem
  for S1,S2,S3 being non empty ManySortedSign, A1 being non-empty
  MSAlgebra over S1, A2 being non-empty MSAlgebra over S2, A3 being MSAlgebra
  over S3 st A1 tolerates A2 & A2 tolerates A3 & A3 tolerates A1 holds A1+*A2
  tolerates A3
proof
  let S1,S2,S3 be non empty ManySortedSign;
  let A1 be non-empty MSAlgebra over S1, A2 be non-empty MSAlgebra over S2, A3
  be MSAlgebra over S3 such that
A1: S1 tolerates S2 and
A2: the Sorts of A1 tolerates the Sorts of A2 and
A3: the Charact of A1 tolerates the Charact of A2 and
A4: S2 tolerates S3 and
A5: the Sorts of A2 tolerates the Sorts of A3 and
A6: the Charact of A2 tolerates the Charact of A3 and
A7: S3 tolerates S1 and
A8: the Sorts of A3 tolerates the Sorts of A1 and
A9: the Charact of A3 tolerates the Charact of A1;
  thus S1+*S2 tolerates S3 by A1,A4,A7,Th3;
  the Sorts of A1+*A2 = (the Sorts of A1)+*the Sorts of A2 by A2,Def4;
  hence the Sorts of A1+*A2 tolerates the Sorts of A3 by A2,A5,A8,FUNCT_4:125;
  the Charact of A1+*A2 = (the Charact of A1)+*the Charact of A2 by A2,Def4;
  hence thesis by A3,A6,A9,FUNCT_4:125;
end;
