reserve a,b,c,d for Ordinal;
reserve l for non empty limit_ordinal Ordinal;
reserve u for Element of l;
reserve A for non empty Ordinal;
reserve e for Element of A;
reserve X,Y,x,y,z for set;
reserve n,m for Nat;
reserve f for Ordinal-Sequence;
reserve U,W for Universe;
reserve F,phi for normal Ordinal-Sequence of W;
reserve g for Ordinal-Sequence-valued Sequence;

theorem
  a-Veblen b in c-Veblen d iff
    a = c & b in d or
    a in c & b in c-Veblen d or
    c in a & a-Veblen b in d
    proof
      hereby
        assume
A1:     a-Veblen b in c-Veblen d;
        per cases by ORDINAL1:14;
        case a = c;
          hence b in d by A1,Th72;
        end;
        case a in c; then
          a-Veblen(c-Veblen d) = c-Veblen d by Th70;
          hence b in c-Veblen d by A1,Th72;
        end;
        case c in a; then
          c-Veblen(a-Veblen b) = a-Veblen b by Th70;
          hence a-Veblen b in d by A1,Th72;
        end;
      end;
      assume
A2:   a = c & b in d or a in c & b in c-Veblen d or c in a & a-Veblen b in d;
      per cases by A2;
      suppose a = c & b in d;
        hence a-Veblen b in c-Veblen d by Th72;
      end;
      suppose
A3:     a in c & b in c-Veblen d; then
        a-Veblen(c-Veblen d) = c-Veblen d by Th70;
        hence a-Veblen b in c-Veblen d by A3,Th72;
      end;
      suppose
A4:     c in a & a-Veblen b in d; then
        c-Veblen(a-Veblen b) = a-Veblen b by Th70;
        hence a-Veblen b in c-Veblen d by A4,Th72;
      end;
    end;
