
theorem
  for x being set, S being non empty strict ManySortedSign
  for A being Boolean MSAlgebra over S st x in the carrier of S holds
  (SingleMSA x) +* A = the MSAlgebra of A
proof
  let x be set, S be non empty strict ManySortedSign;
  let A be Boolean MSAlgebra over S;
  set S1 = SingleMSS x, A1 = SingleMSA x;
  assume
A1: x in the carrier of S;
  then
A2: S1 +* S = S by Th4;
A3: {x} c= the carrier of S by A1,ZFMISC_1:31;
A4: the carrier of S1 = {x} by Def1;
A5: the Sorts of A = (the carrier of S) --> BOOLEAN by CIRCCOMB:57;
  the Sorts of A1 = (the carrier of S1) --> BOOLEAN by CIRCCOMB:57;
  then
A6: the Sorts of A1 tolerates the Sorts of A by A5,FUNCOP_1:87;
A7: the Charact of A = (the Charact of A1)+*the Charact of A;
A8: the Sorts of A1+*A = (the Sorts of A1)+*the Sorts of A by A6,CIRCCOMB:def 4
;
A9: the Charact of A = the Charact of A1+*A by A6,A7,CIRCCOMB:def 4;
A10: dom the Sorts of A1 = the carrier of S1 by PARTFUN1:def 2;
  dom the Sorts of A = the carrier of S by PARTFUN1:def 2;
  hence thesis by A2,A3,A4,A8,A9,A10,FUNCT_4:19;
end;
