reserve X for set;
reserve S for Subset-Family of X;

theorem Thm5:
  for S be cap-finite-partition-closed diff-c=-finite-partition-closed
  Subset-Family of X holds (for A be Element of S, Q be finite Subset of S st
  union Q c= A & Q is a_partition of union Q ex R be finite
  Subset of S st union R misses union Q & Q\/R is a_partition of A)
  proof
    let S be cap-finite-partition-closed diff-c=-finite-partition-closed
    Subset-Family of X;
A1: (for A,B be Element of S holds
    ex P be finite Subset of S st P is a_partition of A/\B) by Lem7;
A2: (for C,D be Element of S st D c= C holds
    ex P be finite Subset of S st P is a_partition of C\D) by Defdiff2;
    let A be Element of S;
    let Q be finite Subset of S;
    assume that
A3: union Q c= A and
A4: Q is a_partition of union Q;
    per cases;
    suppose
A5:   S is empty;
      then
A6:   A is empty by SUBSET_1:def 1;
A7:   union Q misses union {} by ZFMISC_1:2;
A8:   {} is finite Subset of {} &
      {} is a_partition of {} by SUBSET_1:1,EQREL_1:45;
      Q\/{} is a_partition of A by A5,A6,EQREL_1:45;
      hence thesis by A5,A7,A8;
    end;
    suppose
A9:   not S is empty;
      per cases;
      suppose
A10:    A is empty;
A20:    union Q misses union {} by ZFMISC_1:2;
A21:    {} is finite Subset of S &
        {} is a_partition of {} by XBOOLE_1:2,EQREL_1:45;
        Q\/{} is a_partition of A by A4,A10,A3;
        hence thesis by A21,A20;
      end;
      suppose not A is empty;
        hence thesis by A1,A2,A3,A4,A9,Lem6;
      end;
    end;
  end;
