
theorem
  for S1,S2,S3 being non empty ManySortedSign holds (S1+*S2)+*S3 = S1+*(
  S2+*S3 )
proof
  let S1,S2,S3 be non empty ManySortedSign;
  set S12 = S1+*S2, S23 = S2+*S3;
  set S1293 = S12+*S3, S1923 = S1+*S23;
A1: the carrier of S23 = (the carrier of S2) \/ (the carrier of S3) by Def2;
A2: the carrier of S1293 = (the carrier of S12) \/ (the carrier of S3) by Def2;
A3: the carrier of S1923 = (the carrier of S1) \/ (the carrier of S23) by Def2;
  the carrier of S12 = (the carrier of S1) \/ (the carrier of S2) by Def2;
  then
A4: the carrier of S1293 = the carrier of S1923 by A1,A2,A3,XBOOLE_1:4;
A5: the carrier' of S23 = (the carrier' of S2) \/ the carrier' of S3 by Def2;
A6: the carrier' of S1293 = (the carrier' of S12) \/ the carrier' of S3 by Def2
;
A7: the Arity of S23 = (the Arity of S2) +* (the Arity of S3) by Def2;
A8: the Arity of S1923 = (the Arity of S1) +* (the Arity of S23) by Def2;
A9: the carrier' of S1923 = (the carrier' of S1) \/ the carrier' of S23 by Def2
;
  the carrier' of S12 = (the carrier' of S1) \/ the carrier' of S2 by Def2;
  then
A10: the carrier' of S1293 = the carrier' of S1923 by A5,A6,A9,XBOOLE_1:4;
A11: the ResultSort of S1923 = (the ResultSort of S1)+*the ResultSort of S23
  by Def2;
A12: the ResultSort of S1293 = (the ResultSort of S12)+*the ResultSort of S3
  by Def2;
A13: the Arity of S1293 = (the Arity of S12) +* (the Arity of S3) by Def2;
  the Arity of S12 = (the Arity of S1) +* (the Arity of S2) by Def2;
  then
A14: the Arity of S1293 = the Arity of S1923 by A7,A13,A8,FUNCT_4:14;
A15: the ResultSort of S23 = (the ResultSort of S2)+*the ResultSort of S3 by
Def2;
  the ResultSort of S12 = (the ResultSort of S1)+*the ResultSort of S2 by Def2;
  hence thesis by A15,A12,A11,A4,A10,A14,FUNCT_4:14;
end;
