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 Th5:
  for H,H9,f holds f in St(H,E) & f in St(H9,E) iff f in St(H '&' H9,E)
proof
  let H,H9,f;
A1: H '&' H9 is conjunctive;
  then
A2: St(H '&' H9,E) = St(the_left_argument_of(H '&' H9),E) /\
   St(the_right_argument_of(H '&' H9),E) by Lm3;
  H '&' H9 = (the_left_argument_of(H '&' H9)) '&' the_right_argument_of(H
  '&' H9) by A1,ZF_LANG:40;
  then
A3: H = the_left_argument_of(H '&' H9) & H9 = the_right_argument_of(H '&' H9
  ) by ZF_LANG:30;
  hence f in St(H,E) & f in St(H9,E) implies f in St(H '&' H9,E) by A2,
XBOOLE_0:def 4;
  assume f in St(H '&' H9,E);
  hence thesis by A2,A3,XBOOLE_0:def 4;
end;
