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;
