reserve X,Y for set, x,y,z for object, i,j,n for natural number;

theorem Th9:
  for S,E being non void Signature st E is S-extension
  for o being OperSymbol of S
  for a being set for r being Element of S for r1 being Element of E
  st r = r1 & o is_of_type a,r holds o is_of_type a,r1
  proof
    let S,E be non void Signature;
    assume S is Subsignature of E;
    then
A1: the carrier of S c= the carrier of E &
    the carrier' of S c= the carrier' of E &
    the ResultSort of S c= the ResultSort of E &
    the Arity of S c= the Arity of E by INSTALG1:10,11;
    let o be OperSymbol of S;
    let a be set;
    let r be Element of S;
    let r1 be Element of E;
    assume A2: r = r1;
    assume
A3: (the Arity of S).o = a & (the ResultSort of S).o = r;
    dom the Arity of S = the carrier' of S & o in the carrier' of S &
    dom the ResultSort of S = the carrier' of S by FUNCT_2:def 1;
    hence (the Arity of E).o = a & (the ResultSort of E).o = r1
    by A1,A2,A3,GRFUNC_1:2;
  end;
