Pre-Populate based on other inputs

Example.xlsx

Using the attached example, i would like the table on the right (Eqpt + ID#) to pre-populate based on the inputs in columns A-F (at the moment there is no formula, it's all manually entered)
The number of inputs could vary, and may not always be in numeric order of columns A-F (there should only be numeric values entered).
A (TA) up to 24 separate inputs
B (PU) up to 24 separate inputs (either single numbers or pairs)
C (FI) up to 10 separate inputs
D (VE) up to 4 separate inputs
E (LC) up to 2 separate inputs
F (OS) up to 2 separate inputs

I would like to make it so that the table on the right ignores any blanks (as shown in the examples)

An alternative formula approach:
=INDEX($A$1:$F$1,7-MATCH(ROW(A1),SUBTOTAL(3,OFFSET($A$2:$A$7,0,0,6,7-COLUMN($A$1:$F$1))),-1))
in H2 as an array formula (confirm with ctrl+shift+enter)
and
=HLOOKUP(H2,A$1:F$7,COUNTIF($H$1:H1,H2)+2,0)
in I2

then copy down

@Kaper & @ragulduy

Thanks for the replies, i 'll let you know when i've had a chance to test this. It's for work and i've just been given another task to do.

Hi:

Looking at the formula in H2 first:
=INDEX($A$1:$F$1,7-MATCH(ROW(A1),SUBTOTAL(3,OFFSET($A$2:$A$7,0,0,6,7-COLUMN($A$1:$F$1))),-1))

A1:F1 is to do with column headers, index(a,b) finds the bth term of the range a. So the 7-Match part is to do with columns. As Is the 7-column(A1:F1) part. The two parts relating to rows are in the offset function, the A2:A7 is the initial range and 6 is the number of rows, so for example, if your range went from A2:A14, the formula would become:
=INDEX($A$1:$F$1,7-MATCH(ROW(A1),SUBTOTAL(3,OFFSET($A$2:$A$14,0,0,13,7-COLUMN($A$1:$F$1))),-1))
Where A2:A14 is the range and 13 is the number of rows in the range.

The holookup is a bit easier, you just need to change the lookup range, so for 14 rows
=HLOOKUP(H2,A$1:F$8,COUNTIF($H$1:H1,H2)+2,0)
would become
=HLOOKUP(H2,A$1:F$14,COUNTIF($H$1:H1,H2)+2,0)

Unforunately, this approach is based on your data being in a table on one sheet as you have in your first workbook. Once each heading is moved onto different sheets, I don't think it would be possible to adapt it. You would need to build the table from the data on the separate worksheets first.

I suppose I could use a hidden part of the sheet to build the data from the separate ones onto a single page and then use your formula.

Thanks for the help.

EDIT:
sorry for the double post, something strange happened to my browser.