reserve a,a1,a2,b,c,d for Ordinal,
  n,m,k for Nat,
  x,y,z,t,X,Y,Z for set;

theorem Th6:
  a\b <> {} implies inf(a\b) = b & sup(a\b) = a & union(a\b) = union a
  proof assume
A1: a\b <> {};
    set x = the Element of a\b;
A2: x in a\b by A1;
A3: x in a & x nin b by A1,XBOOLE_0:def 5; then
A:  b c= x by ORDINAL6:4; 
    not b in b; then
    b in a & b nin b by A,A3,ORDINAL1:12; then
A4: b in a\b by XBOOLE_0:def 5;
    hence inf(a\b) c= b by ORDINAL2:14;
    inf(a\b) in a\b by A2,ORDINAL2:17; then
    inf(a\b) nin b by XBOOLE_0:def 5;
    hence b c= inf(a\b) by ORDINAL6:4;
A5: On(a\b) = a\b by ORDINAL6:2;
    thus sup(a\b) c= a by A5,ORDINAL2:def 3;
    thus a c= sup(a\b)
    proof
      let c; assume
A6:   c in a;
A7:   b in sup(a\b) by A4,ORDINAL2:19;
      per cases;
      suppose c in b;
        hence thesis by A7,ORDINAL1:10;
      end;
      suppose c nin b; then
        c in a\b by A6,XBOOLE_0:def 5;
        hence thesis by ORDINAL2:19;
      end;
    end;
    thus union(a\b) c= union a by ZFMISC_1:77;
    for y be object st y in union a holds y in union(a\b)
    proof
      let y be object; assume y in union a; then
      consider x such that
A8:   y in x & x in a by TARSKI:def 4;
      reconsider x as Ordinal by A8;
      per cases by ORDINAL6:4;
      suppose x in b; then
        y in b by A8,ORDINAL1:10;
        hence thesis by A4,TARSKI:def 4;
      end;
      suppose b c= x; then
        x in a\b by A8,ORDINAL6:5;
        hence thesis by A8,TARSKI:def 4;
      end;
    end;
    hence union a c= union(a\b);
  end;
