reserve fi,psi for Ordinal-Sequence,
  A,A1,B,C,D for Ordinal,
  X,Y for set,
  x,y for object;

theorem
  A-^(B+^C) = (A-^B)-^C
proof
  now
    per cases;
    suppose
      B+^C c= A;
      then A = B+^C+^(A-^(B+^C)) by Def5;
      then A = B+^(C+^(A-^(B+^C))) by Th30;
      then C+^(A-^(B+^C)) = A-^B by Th52;
      hence thesis by Th52;
    end;
    suppose
A1:   not B+^C c= A;
A2:   now
        assume A = B+^(A-^B);
        then not C c= A-^B by A1,ORDINAL2:33;
        hence A-^B-^C = {} by Def5;
      end;
      B c= A or not B c= A;
      then
A3:   A = B+^(A-^B) or A-^B = {} by Def5;
      A-^(B+^C) = {} by A1,Def5;
      hence thesis by A3,A2,Th56;
    end;
  end;
  hence thesis;
end;
