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

theorem Th4:
  mi (A ^ B) = A implies for a be set st a in A ex b be set st b in B & b c= a
proof
  assume
A1: mi (A ^ B) = A;
  let a be set;
A2: mi (A ^ B) c= A ^ B by SUBSTLAT:8;
  assume a in A;
  then consider b,c be set such that
  b in A and
A3: c in B and
A4: a = b \/ c by A1,A2,SUBSTLAT:15;
  take c;
  thus thesis by A3,A4,XBOOLE_1:7;
end;
