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

theorem Th7:
  a\b is non empty finite implies
  ex n being Nat st a = b+^n
  proof assume
A1: a\b is non empty finite;
    set x = the Element of a\b;
A2: x in a & x nin b by A1,XBOOLE_0:def 5; then
    b c= x by ORDINAL6:4; then
    consider c such that
A3: a = b+^c & c <> {} by A2,ORDINAL1:12,ORDINAL3:28;
    deffunc F(Ordinal) = b+^$1;
    consider f being Sequence such that
A4: dom f = omega & for d st d in omega holds f.d = F(d) from ORDINAL2:sch 2;
    f is one-to-one
    proof
      let x,y be object; assume
A5:   x in dom f & y in dom f & f.x = f.y & x <> y; then
      reconsider x,y as Element of omega by A4;
A6:   f.x = F(x) & f.y = F(y) by A4;
      x in y or y in x by A5,ORDINAL1:14; then
      b+^x in b+^y or b+^y in b+^x by ORDINAL2:32;
      hence contradiction by A5,A6;
    end; then
A7: omega, rng f are_equipotent by A4;
    now assume
      omega c= c; then
A8:   F(omega) c= a by A3,ORDINAL2:33;
      rng f c= a\b
      proof
        let y be object; assume y in rng f; then
        consider x being object such that
A9:     x in dom f & y = f.x by FUNCT_1:def 3;
        reconsider x as Element of omega by A4,A9;
A10:     y = F(x) by A4,A9;
        b c= F(x) & F(x) in F(omega) by ORDINAL2:32,ORDINAL3:24; then
        y nin b & y in a by A8,A10,ORDINAL6:4;
        hence thesis by XBOOLE_0:def 5;
      end;
      hence contradiction by A1,A7,CARD_1:38;
    end; then
    c in omega by ORDINAL6:4;
    hence thesis by A3;
  end;
