theorem
  for f being empty valuation of C
  holds f# = id Union the Sorts of Free(C, MSVars C)
proof
  let f be empty valuation of C;
A1: dom (f# ) = Union the Sorts of Free(C, MSVars C) by FUNCT_2:def 1;
  now
    let x be object;
    assume x in Union the Sorts of Free(C, MSVars C);
    then reconsider t = x as expression of C;
    thus (f# ).x = t at f .= x by Th139;
  end;
  hence thesis by A1,FUNCT_1:17;
end;
