reserve k,m,n for Element of NAT,
  a,X,Y for set,
  D,D1,D2 for non empty set;
reserve p,q for FinSequence of NAT;
reserve x,y,z,t for Variable;
reserve F,F1,G,G1,H,H1 for ZF-formula;
reserve sq,sq9 for FinSequence;
reserve L,L9 for FinSequence;
reserve j,j1 for Element of NAT;

theorem Th83:
  Subformulae (H '&' F) = Subformulae H \/ Subformulae F \/ { H '&' F }
proof
  now
    let a be object;
A1: a in Subformulae H \/ Subformulae F implies a in Subformulae H or a
    in Subformulae F by XBOOLE_0:def 3;
    thus a in Subformulae H '&' F implies a in Subformulae H \/ Subformulae F
    \/ { H '&' F }
    proof
      assume a in Subformulae H '&' F;
      then consider G such that
A2:   G = a and
A3:   G is_subformula_of H '&' F by Def42;
      now
        assume G <> H '&' F;
        then G is_proper_subformula_of H '&' F by A3;
        then G is_subformula_of H or G is_subformula_of F by Th70;
        then a in Subformulae H or a in Subformulae F by A2,Def42;
        hence a in Subformulae H \/ Subformulae F by XBOOLE_0:def 3;
      end;
      then a in Subformulae H \/ Subformulae F or a in { H '&' F } by A2,
TARSKI:def 1;
      hence thesis by XBOOLE_0:def 3;
    end;
A4: now
      assume a in { H '&' F };
      then
A5:   a = H '&' F by TARSKI:def 1;
      H '&' F is_subformula_of H '&' F by Th59;
      hence a in Subformulae H '&' F by A5,Def42;
    end;
A6: now
      assume a in Subformulae F;
      then consider G such that
A7:   G = a and
A8:   G is_subformula_of F by Def42;
      F is_immediate_constituent_of H '&' F;
      then F is_proper_subformula_of H '&' F by Th61;
      then F is_subformula_of H '&' F;
      then G is_subformula_of H '&' F by A8,Th65;
      hence a in Subformulae H '&' F by A7,Def42;
    end;
A9: now
      assume a in Subformulae H;
      then consider G such that
A10:  G = a and
A11:  G is_subformula_of H by Def42;
      H is_immediate_constituent_of H '&' F;
      then H is_proper_subformula_of H '&' F by Th61;
      then H is_subformula_of H '&' F;
      then G is_subformula_of H '&' F by A11,Th65;
      hence a in Subformulae H '&' F by A10,Def42;
    end;
    assume a in Subformulae H \/ Subformulae F \/ { H '&' F };
    hence a in Subformulae H '&' F by A1,A9,A6,A4,XBOOLE_0:def 3;
  end;
  hence thesis by TARSKI:2;
end;
