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

theorem Th6:
  for D be non empty set, F be PartFunc of D,REAL, r,s be Real
 st r <> 0 holds F"{s/r} = (r(#)F)"{s}
proof
  let D be non empty set, F be PartFunc of D,REAL, r,s be Real;
  assume
A1: r <> 0;
  thus F"{s/r} c= (r(#)F)"{s}
  proof
    let x be object;
    assume
A2: x in F"{s/r};
    then reconsider d=x as Element of D;
    d in dom F by A2,FUNCT_1:def 7;
    then
A3: d in dom(r(#)F) by VALUED_1:def 5;
    F.d in {s/r} by A2,FUNCT_1:def 7;
    then F.d = s/r by TARSKI:def 1;
    then r*F.d = s by A1,XCMPLX_1:87;
    then (r(#)F).d = s by A3,VALUED_1:def 5;
    then (r(#)F).d in {s} by TARSKI:def 1;
    hence thesis by A3,FUNCT_1:def 7;
  end;
  let x be object;
  assume
A4: x in (r(#)F)"{s};
  then reconsider d=x as Element of D;
A5: d in dom(r(#)F) by A4,FUNCT_1:def 7;
  (r(#)F).d in {s} by A4,FUNCT_1:def 7;
  then (r(#)F).d = s by TARSKI:def 1;
  then r*F.d = s by A5,VALUED_1:def 5;
  then F.d = s/r by A1,XCMPLX_1:89;
  then
A6: F.d in {s/r} by TARSKI:def 1;
  d in dom F by A5,VALUED_1:def 5;
  hence thesis by A6,FUNCT_1:def 7;
end;
