reserve F,H,H9 for ZF-formula,
  x,y,z,t for Variable,
  a,b,c,d,A,X for set;
reserve E for non empty set,
  f,g,h for Function of VAR,E,
  v1,v2,v3,v4,v5,u5 for Element of VAL E;

theorem Th4:
  for H,f holds not f in St(H,E) iff f in St('not' H,E)
proof
  let H,f;
A1: 'not' H is negative;
  then H = the_argument_of 'not' H by ZF_LANG:def 30;
  then
A2: St('not' H,E) = (VAL E) \ St(H,E) by A1,Lm3;
  f in VAL E by FUNCT_2:8;
  hence not f in St(H,E) implies f in St('not' H,E) by A2,XBOOLE_0:def 5;
  assume f in St('not' H,E);
  hence thesis by A2,XBOOLE_0:def 5;
end;
