reserve x,y,x1,x2,z for set,
  n,m,k for Nat,
  t1 for (DecoratedTree of [: NAT,NAT :]),
  w,s,t,u for FinSequence of NAT,
  D for non empty set;
reserve s9,w9,v9 for Element of NAT*;
reserve p,q for MP-variable;
reserve A,A1,B,B1,C,C1 for MP-wff;

theorem Th37:
  A = VERUM or (ex p st A = @p) or (ex B st A = 'not' B) or (ex B
  st A = (#) B) or ex B,C st A = B '&' C
proof
  now
    per cases by NAT_1:25;
    suppose
      card dom A = 1;
      hence thesis by Th31;
    end;
    suppose
      card dom A > 1;
      then card dom A >= 1+1 by NAT_1:13;
      hence thesis by Th32;
    end;
  end;
  hence thesis;
end;
