reserve n,k for Nat,
  X for non empty set,
  S for SigmaField of X;

theorem
  for X,Y be non empty set, E be set, F,G be Function of X,Y st
  for x be Element of X holds G.x = E \ F.x holds
    union rng G = E \ meet rng F
proof
  let X,Y be non empty set,E be set,F,G be Function of X,Y;
  assume
A1: for x be Element of X holds G.x = E \ F.x;
A2: dom G = X by FUNCT_2:def 1;
  now
    let Z be object;
    assume Z in DIFFERENCE({E},rng F);
    then consider X1,Y be set such that
A3: X1 in {E} and
A4: Y in rng F and
A5: Z = X1 \ Y by SETFAM_1:def 6;
    consider x be object such that
A6: x in dom F and
A7: Y = F.x by A4,FUNCT_1:def 3;
    reconsider x as Element of X by A6;
    X1 = E by A3,TARSKI:def 1;
    then Z = G.x by A1,A5,A7;
    hence Z in rng G by A2,FUNCT_1:3;
  end;
  then
A8: DIFFERENCE({E},rng F) c= rng G;
A9: dom F = X by FUNCT_2:def 1;
  now
    let Z be object;
A10: E in {E} by TARSKI:def 1;
    assume Z in rng G;
    then consider x be object such that
A11: x in dom G and
A12: Z = G.x by FUNCT_1:def 3;
    reconsider x as Element of X by A11;
A13: F.x in rng F by A9,FUNCT_1:3;
    Z = E \ F.x by A1,A12;
    hence Z in DIFFERENCE({E},rng F) by A13,A10,SETFAM_1:def 6;
  end;
  then rng G c= DIFFERENCE({E},rng F);
  then DIFFERENCE({E},rng F) = rng G by A8,XBOOLE_0:def 10;
  hence thesis by SETFAM_1:27;
end;
