theorem
  for X being Subset of Vars
  holds (C idval X)# = id Union the Sorts of Free(C, MSVars C)
proof
  let X be Subset of Vars;
  set f = C idval X;
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 Th137;
  end;
  hence thesis by A1,FUNCT_1:17;
end;
