
theorem Th4:
  for x being set, S being non empty ManySortedSign st x in the carrier of S
  holds (SingleMSS x) +* S = the ManySortedSign of S
proof
  let x be set, S be non empty ManySortedSign;
  set T = (SingleMSS x) +* S;
  assume x in the carrier of S;
  then {x} c= the carrier of S by ZFMISC_1:31;
  then
A1: {x} \/ the carrier of S = the carrier of S by XBOOLE_1:12;
A2: {} \/ the carrier' of S = the carrier' of S;
A3: the carrier of SingleMSS x = {x} by Def1;
A4: the ResultSort of SingleMSS x = {};
A5: the Arity of SingleMSS x = {};
A6: {}+*the ResultSort of S = the ResultSort of S;
A7: {}+*the Arity of S = the Arity of S;
A8: the carrier of T = the carrier of S by A1,A3,CIRCCOMB:def 2;
A9: the carrier' of T = the carrier' of S by A2,A4,CIRCCOMB:def 2;
  the ResultSort of T = the ResultSort of S by A4,A6,CIRCCOMB:def 2;
  hence thesis by A5,A7,A8,A9,CIRCCOMB:def 2;
end;
