
theorem
  for S1 being feasible ManySortedSign for S2 being Subsignature of S1
st S1 is Subsignature of S2 holds the ManySortedSign of S1 = the ManySortedSign
  of S2
proof
  let S1 be feasible ManySortedSign;
  let S2 be Subsignature of S1;
A1: the carrier of S2 c= the carrier of S1 by Th10;
  assume
A2: S1 is Subsignature of S2;
  then the carrier of S1 c= the carrier of S2 by Th10;
  then
A3: the carrier of S1 = the carrier of S2 by A1,XBOOLE_0:def 10;
A4: the carrier' of S2 c= the carrier' of S1 by Th10;
  the carrier' of S1 c= the carrier' of S2 by A2,Th10;
  then
A5: the carrier' of S1 = the carrier' of S2 by A4,XBOOLE_0:def 10;
A6: the Arity of S2 c= the Arity of S1 by Th11;
A7: the ResultSort of S2 c= the ResultSort of S1 by Th11;
  the ResultSort of S1 c= the ResultSort of S2 by A2,Th11;
  then
A8: the ResultSort of S1 = the ResultSort of S2 by A7,XBOOLE_0:def 10;
  the Arity of S1 c= the Arity of S2 by A2,Th11;
  hence thesis by A6,A3,A5,A8,XBOOLE_0:def 10;
end;
