reserve i for Nat,
  j for Element of NAT,
  X,Y,x,y,z for set;
reserve C for initialized ConstructorSignature,
  s for SortSymbol of C,
  o for OperSymbol of C,
  c for constructor OperSymbol of C;
reserve a,b for expression of C, an_Adj C;
reserve t, t1,t2 for expression of C, a_Type C;
reserve p for FinSequence of QuasiTerms C;
reserve e for expression of C;
reserve a,a9 for expression of C, an_Adj C;
reserve q for pure expression of C, a_Type C,
  A for finite Subset of QuasiAdjs C;
reserve T for quasi-type of C;

theorem Th131:
  for X being Subset of Vars holds dom (C idval X) = X &
  for x being variable st x in X holds (C idval X).x = x-term C
proof
  let X be Subset of Vars;
  set f = C idval X;
  thus dom f c= X
  proof
    let a being object;
    assume a in dom f;
    then [a,f.a] in f by FUNCT_1:def 2;
    then ex x being variable st [a,f.a] = [x,x-term C] & x in X;
    hence thesis by XTUPLE_0:1;
  end;
  hereby
    let x be object;
    assume
A1: x in X;
    then reconsider a = x as variable;
    [a,a-term C] in f by A1;
    hence x in dom f by FUNCT_1:1;
  end;
  let x be variable;
  assume x in X;
  then [x,x-term C] in C idval X;
  hence thesis by FUNCT_1:1;
end;
