suppose {A,B,C,D} = {3,4,6,9} and {E,F,G,H} = {4,5,8,9}
and

 ABCD
-EFGH
-----
 3487

We try for the 4th (right-most) column (D-H=7) with each of the digits 
for the larger number.

D    3  4  6  9  <--possible digits of the larger number
H    6  7  9  2  <--digits required to get 7 in the bottom for the 4th digit
-    -  -  -  -
7    7  7  7  7


Only one of those digits for H is one of the digits for the smaller
number, so the problem now becomes this, with D=6 and H=9

 ABC6
-EFG9
-----
 3487  

So we have to borrow one from the C.
Now we try for the 3rd (next to right-most) column (C-G=8) with each of the
remaining digits with the larger number.  Don't forget that we borrowed 1 from
the C.

C      3  4  9  <--possible digits of the larger number for C

C-1    2  3  5  <--possible digits after borrowing 1 from C
  G    3  4  6  <--digits required to get 9 in the bottom for the 3rd digit
  -    -  -  -  
  9    9  9  9  

As before, only one of those digits for G is one of the digits for the smaller
number, so the problem now becomes this with C=4 and G=4:

 AB46
-EF49
-----
 3497 

So we have to borrow one from the B.
Now we try for the 3rd (next to right-most) column (B-F=4) with each of the
remaining digits with the larger number.  Don't forget that we borrowed 1 from the B.

  B    3  9  <--possible digits of the larger number for B

B-1    2  5  <--possible digits after borrowing 1 from B
  F    8  1  <--digits required to get 4 in the bottom for the 2nd digit
  -    -  -    
  4    4  4   

As before, only one of those digits for F is one of the digits for the smaller
number, so the problem now becomes this, with B=3 and F=8

 A346
-E849
-----
 3497 

Now the only thing left for A and E are 9 and 5 respectively.  So

 9346          
-5849
-----          
 3497

is the only possibility.  The larger number is 9346.

Edwin