theorem Th7:
  rng L = Subformulae H & L is one-to-one & (not F in W) & W1 = W
  \/ {F} implies len(L,W1) = len(L,W) + len F
proof
  assume that
A1: rng L = Subformulae H & L is one-to-one and
A2: not F in W and
A3: W1 = W \/ {F};
A4: for x being object holds x in W1\{F} implies x in W
  proof let x be object;
    assume x in W1\{F};
    then x in W1 & not x in {F} by XBOOLE_0:def 5;
    hence thesis by A3,XBOOLE_0:def 3;
  end;
  for x being object holds x in W implies x in W1\{F}
  proof let x be object;
    assume x in W;
    then (not x in {F})& x in W1 by A2,A3,TARSKI:def 1,XBOOLE_0:def 3;
    hence thesis by XBOOLE_0:def 5;
  end;
  then
A5: W1\{F} = W by A4,TARSKI:2;
  F in {F} by TARSKI:def 1;
  then F in W1 by A3,XBOOLE_0:def 3;
  then len(L,W) = len(L,W1) - len F by A1,A5,Th6;
  hence thesis;
end;
