reserve V, C, x, a, b for set;
reserve A, B for Element of SubstitutionSet (V, C);

theorem Th5:
  (for a be set st a in A ex b be set st b in B & b c= a) implies
  mi (A ^ B) = A
proof
  assume
A1: for a be set st a in A ex b be set st b in B & b c= a;
A2: mi (A ^ B) c= A ^ B by SUBSTLAT:8;
  thus mi (A ^ B) c= A
  proof
    let a be object;
     reconsider aa=a as set by TARSKI:1;
A3: A c= PFuncs (V, C) by FINSUB_1:def 5;
    assume
A4: a in mi (A ^ B);
    then consider b, c be set such that
A5: b in A and
    c in B and
A6: a = b \/ c by A2,SUBSTLAT:15;
A7: b c= aa by A6,XBOOLE_1:7;
    consider b1 be set such that
A8: b1 in B and
A9: b1 c= b by A1,A5;
    B c= PFuncs (V, C) by FINSUB_1:def 5;
    then reconsider b9 = b, b19 = b1 as Element of PFuncs (V, C) by A5,A8,A3;
A10: b = b1 \/ b by A9,XBOOLE_1:12;
    b19 tolerates b9 by A9,PARTFUN1:54;
    then b in A ^ B by A5,A8,A10,SUBSTLAT:16;
    hence thesis by A4,A5,A7,SUBSTLAT:6;
  end;
  thus A c= mi (A ^ B)
  proof
    let a be object;
     reconsider aa=a as set by TARSKI:1;
A11: A c= PFuncs (V, C) by FINSUB_1:def 5;
    assume
A12: a in A;
    then consider b be set such that
A13: b in B and
A14: b c= aa by A1;
    B c= PFuncs (V, C) by FINSUB_1:def 5;
    then reconsider a9 = a, b9 = b as Element of PFuncs (V, C) by A12,A13,A11;
A15: a9 tolerates b9 by A14,PARTFUN1:54;
A16: a in mi A by A12,SUBSTLAT:11;
A17: for b be finite set st b in A ^ B & b c= aa holds b = a
    proof
      let b be finite set;
      assume that
A18:  b in A ^ B and
A19:  b c= aa;
      consider c, d be set such that
A20:  c in A and
      d in B and
A21:  b = c \/ d by A18,SUBSTLAT:15;
      c c= b by A21,XBOOLE_1:7;
      then c c= aa by A19;
      then c = a by A16,A20,SUBSTLAT:6;
      hence thesis by A19,A21,Lm1;
    end;
    a9 = a9 \/ b9 by A14,XBOOLE_1:12;
    then
A22: a9 in A ^ B by A12,A13,A15,SUBSTLAT:16;
    aa is finite by A12,Th1;
    hence thesis by A22,A17,SUBSTLAT:7;
  end;
end;
