reserve n,m for Nat,
  r,r1,r2,s,t for Real,
  x,y for set;

theorem Th50:
  for D be non empty set, F be PartFunc of D,REAL, r,s
  holds F"{s+r} = (F-r)"{s}
proof
  let D be non empty set, F be PartFunc of D,REAL, r,s;
  thus F"{s+r} c= (F-r)"{s}
  proof
    let x be object;
    assume
A1: x in F"{s+r};
    then reconsider d=x as Element of D;
A2: d in dom F by A1,FUNCT_1:def 7;
    F.d in {s+r} by A1,FUNCT_1:def 7;
    then F.d = s+r by TARSKI:def 1;
    then F.d - r = s;
    then (F-r).d = s by A2,VALUED_1:3;
    then
A3: (F-r).d in {s} by TARSKI:def 1;
    d in dom(F-r) by A2,VALUED_1:3;
    hence thesis by A3,FUNCT_1:def 7;
  end;
  let x be object;
  assume
A4: x in (F-r)"{s};
  then reconsider d=x as Element of D;
  d in dom(F-r) by A4,FUNCT_1:def 7;
  then
A5: d in dom F by VALUED_1:3;
  (F-r).d in {s} by A4,FUNCT_1:def 7;
  then (F-r).d = s by TARSKI:def 1;
  then F.d -r= s by A5,VALUED_1:3;
  then F.d in {s+r} by TARSKI:def 1;
  hence thesis by A5,FUNCT_1:def 7;
end;
