Here's the full extract from my solver log for the first move, after which, as

angusj points out, the puzzle is straightforward to complete. You might find it useful because it attempts to describe the logic from first principles and avoids specialist terms (apart from the chain notation, of course) such as 'Colouring' and 'Swordfish'. The downside is that more experienced users might find it frustratingly verbose. When I write out a chain, I use the symbol '-' to indicate a strong link (i.e. exactly two candidates for that cell/row/column/box) or '~' for a weak link (i.e. more than two candidates). I believe

wolfgang introduced this notation to the forum:

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`Consider the chain r9c2-7-r5c2-7-r5c8-7-r6c9~7~r9c9.`

When the cell r9c9 contains the value 7, so does the cell r9c2 - a contradiction.

Therefore, the cell r9c9 cannot contain the value 7.

- The move r9c9:=7 has been eliminated.

Consider the chain r1c9-7-r1c6~7~r7c6-7-r7c1-7-r6c1-7-r6c9.

The cell r6c9 must contain the value 7 if the cell r1c9 doesn't.

Therefore, these two cells are the only candidates for the value 7 in Column 9.

- The move r2c9:=7 has been eliminated.

Consider the chain r1c6-7-r1c9-7-r6c9-7-r6c1-7-r7c1-7-r7c6.

The cell r7c6 must contain the value 7 if the cell r1c6 doesn't.

Therefore, these two cells are the only candidates for the value 7 in Column 6.

- The move r2c6:=7 has been eliminated.

Consider the chains r5c8-3-r3c8~5~r8c8 and r5c8-3-r5c3-5-r8c3.

Whichever of the 2 candidates in Row 8 contains the value 5, the cell r5c8 does not contain the value 3.

- The move r5c8:=3 has been eliminated.

The value 7 is the only candidate for the cell r5c8.